By Theorem REF , Theorem REF  and Theorem REF  we have that \(\hat{\phi }\)  is intrinsic differentiable a.e. in \(\hat{ \mathcal {O}}\)  and for all \(i(a)\in \hat{ \mathcal {O}} \)  point of intrinsic differentiability of \(\hat{\phi }\) , there exists a unique \(d\hat{\phi }_{i(a)}:\mathbb {W}\rightarrow  intrinsic differential of \)\( at \) i(a)\( such that\begin{equation*}\mathrm {graph}\,(d\hat{\phi }_{i(a)})=\Big \lbrace  (p_1,\dots , p_{m+n}) \in \mathbb {G}\, : \, \sum _{j=1}^{m} \nu _{\mathcal {E}}^{(j)} (\Phi (a)) p_j =0 \Big \rbrace \end{equation*}where \) E(1),..., E(m)\( are the components of \) E\(. Then we obtain\begin{equation*}\begin{aligned}\left\lbrace  b\cdot d\hat{\phi }_{i(a)}(b) \in \mathbb {G}\, :\, b\in \mathbb {W}\right\rbrace  =\Big \lbrace  (p_1,\dots , p_{m+n}) \in \mathbb {G}\, : \, \sum _{j=1}^{m} \nu _{\mathcal {E}}^{(j)} (\Phi (a)) p_j =0 \Big \rbrace .\end{aligned}\end{equation*}By \) (REF )\( there is \) ( D2  (a), ..., Dm  (a) )Rm-1\( associated to \) di(a)\( such that\)\(d\hat{\phi }_{i(a)}(b)= \Big ( \sum _{j=2}^{m} D_j^{\phi } \phi (a) x_j ,0,\dots , 0\Big ),\)\(for all \) b=(0,x2,..., xm, y1,..., yn)W\(, and consequently recalling that \) (bdi(a)(b))1=( j=2m Dj (a) xj, x2,...,xm )\( we deduce\begin{equation*}\nu _{\mathcal {E}}^{(1)} (\Phi (a)) \sum _{j=2}^{m} D_j^{\phi } \phi (a) x_j + \sum _{j=2}^{m} \nu _{\mathcal {E}}^{(j)} (\Phi (a)) x_j =0\end{equation*}for all \) b=(0,x2,..., xm, y1,..., yn)W\(. The thesis follows choosing \) b=(0,..., 0 ,1,0,..., 0)\( where \) j\(-th element is \) 1\( for \) j=2,...,m\( and recalling that \) | E((a))|Rm=1\( a.e. in \) O \(.\)
Lemma 5.2 ([1]}, Lemma 4.2.10) 
Let \(\hat{\phi }:\hat{ \mathcal {O}} \rightarrow  be an intrinsic Lipschitz function. Then there exists \) C(W, >0\( such that\begin{equation*}(\Phi )_* (\mathcal {L}^{m+n-1} \mathop {\hbox{\vrule height 7pt width .5pt depth 0pt\vrule height .5pt width 6pt depth 0pt}}\nolimits \mathbb {W})=- C(\mathbb {W},  \nu _{\mathcal {E}}^{(1)} |\partial \mathcal {E}|_\mathbb {G}\end{equation*}where \) ()* (Lm+n-1 W)\( denotes the image of \) Lm+n-1 W\( under the map \)\( and \) E(1)\( is the first component of the intrinsic generalized inward normal to the intrinsic subgraph \) E\(. \)
Proposition 5.3 
Let \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 and \(, \) W\( the complementary subgroups defined in (\ref {5.2.0}). Let \) : O  be an intrinsic Lipschitz function, where \(\hat{ \mathcal {O}}\)  is an open subset of \(\mathbb {W}\)  and \(\phi :\mathcal {O} \rightarrow \mathbb {R}\)  is the map associated to \(\hat{\phi }\)  as in \((\ref {phipsi})\) .
Then for every \(\xi =(\xi _1,\dots , \xi _m) \in 1_c (\hat{ \mathcal {O}} \cdot  \mathbb {R}^m)\)  we have
\(\int _{\mathcal {E}} \mbox{div} _\mathbb {G}\xi \, d\mathcal {L}^{m+n} = \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \xi _1 \circ \Phi - \sum _{j=2}^{m} D_j^{\phi } \phi ( \xi \circ \Phi )_j \, d\mathcal {L}^{m+n-1}\) 
where \(C(\mathbb {W},>0\)  is given by Lemma \(\ref {coro4.3}\)  and \(\Phi :\mathcal {O} \rightarrow \mathbb {G}\)  is the map graph of \(\hat{\phi }\) .

Thanks to Proposition REF  we know that \(\mathcal {E}\)  is a set of locally finite perimeter in \(\mathbb {G}\)  and consequently by Structure Theorem of BV\(_\mathbb {G}\)  functions (see Theorem REF ) there exists a unique \(|\partial \mathcal {E}|_\mathbb {G}\) -measurable function \(\nu _\mathcal {E}:\hat{ \mathcal {O}} \cdot \mathbb {R}^m\)  such that \(|\nu _\mathcal {E}|_{\mathbb {R}^{m}}=1\)  \(|\partial \mathcal {E}|_\mathbb {G}\) -a.e. in \(\hat{ \mathcal {O}} \cdot  and\begin{equation}\int _{\mathcal {E}} \mbox{div} _\mathbb {G}\xi \, d\mathcal {L}^{m+n} = - \int _{\hat{ \mathcal {O}} \cdot  \langle \xi ,\nu _\mathcal {E}\rangle \, d|\partial \mathcal {E}|_\mathbb {G}}for all \xi \in 1_c (\hat{ \mathcal {O}} \cdot  \mathbb {R}^m) with |\xi |_{\mathbb {R}^{m}}\le 1.Now using the Lemma \ref {coro4.2} we have that the first component \nu _\mathcal {E}^{(1)} of \nu _\mathcal {E} is such that \nu _\mathcal {E}^{(1)}\ne 0 |\partial \mathcal {E}|_\mathbb {G}-a.e. in \hat{ \mathcal {O}} \cdot . Moreover from Lemma \ref {coro4.3} there exists C(\mathbb {W},>0 such that\begin{equation*}-\int _{\hat{ \mathcal {O}} \cdot  \langle \xi ,\nu _\mathcal {E}\rangle \, d|\partial \mathcal {E}|_\mathbb {G}=- \int _{\hat{ \mathcal {O}} \cdot  \frac{\langle \xi ,\nu _\mathcal {E}\rangle }{\nu _\mathcal {E}^{(1)}}\, \nu _\mathcal {E}^{(1)} \, d|\partial \mathcal {E}|_\mathbb {G}= \frac{1}{C(\mathbb {W},} \int _{\hat{ \mathcal {O}} \cdot  \frac{\langle \xi ,\nu _\mathcal {E}\rangle }{\nu _\mathcal {E}^{(1)}}\, d\Phi _* (\mathcal {L}^{m+n-1} \mathop {\hbox{\vrule height 7pt width .5pt depth 0pt\vrule height .5pt width 6pt depth 0pt}}\nolimits \mathbb {W}).}Finally since a change of variables we conclude that\begin{equation*}\frac{1}{C(\mathbb {W},} \int _{\hat{ \mathcal {O}} \cdot  \frac{ \langle \xi ,\nu _\mathcal {E}\rangle }{\nu _\mathcal {E}^{(1)}}\, d\Phi _* (\mathcal {L}^{m+n-1} \mathop {\hbox{\vrule height 7pt width .5pt depth 0pt\vrule height .5pt width 6pt depth 0pt}}\nolimits \mathbb {W}) = \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \frac{\langle \nu _\mathcal {E}\circ \Phi , \xi \circ \Phi \rangle }{\nu _\mathcal {E}^{(1)} \circ \Phi } \, d\mathcal {L}^{m+n-1}}and consequently, by (\ref {63cit}) for all \xi =(\xi _1,\dots , \xi _m)\in 1_c (\hat{ \mathcal {O}} \cdot  \mathbb {R}^m) with |\xi |_{\mathbb {R}^{m}}\le 1\begin{equation}\begin{aligned}\int _{\mathcal {E}} \mbox{div} _\mathbb {G}\xi \, d\mathcal {L}^{m+n} & = -\int _{\hat{ \mathcal {O}} \cdot  \langle \xi ,\nu _\mathcal {E}\rangle \, d|\partial \mathcal {E}|_\mathbb {G}\\& = \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \frac{\langle \nu _\mathcal {E}\circ \Phi , \xi \circ \Phi \rangle }{\nu _\mathcal {E}^{(1)} \circ \Phi } \, d\mathcal {L}^{m+n-1}\\& = \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \biggl ( \xi _1 \circ \Phi + \sum _{j=2}^{m} \frac{(\nu _\mathcal {E}\circ \Phi )_j}{\nu _\mathcal {E}^{(1)} \circ \Phi } \, (\xi \circ \Phi )_j \biggl )\, d\mathcal {L}^{m+n-1}\\& = \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \biggl ( \xi _1 \circ \Phi - \sum _{j=2}^{m} D_j^{\phi } \phi ( \xi \circ \Phi )_j \biggl )\, d\mathcal {L}^{m+n-1}}\end{aligned}Hence putting together the last equality and (\ref {eqDIV}) we obtain the thesis.\end{equation}\end{equation*}\begin{prop}Let \mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda ) be a Carnot group of step 2 and , \mathbb {W} the complementary subgroups defined in (\ref {5.2.0}). Let \hat{\phi }:\hat{ \mathcal {O}} \rightarrow  be a locally intrinsic Lipschitz function, where \hat{ \mathcal {O}} is an open subset of \mathbb {W} and \phi :\mathcal {O} \rightarrow \mathbb {R} is the map associated to \hat{\phi } as in (\ref {phipsi}).Then the intrinsic gradient (D_2^\phi \phi ,\dots ,D_m^\phi \phi ) is also distributional, i.e.\begin{equation*}\int _\mathcal {O} \phi \left( \,X_j \zeta +\phi \sum _{s=1}^n b_ {j1 }^{(s)} Y_s \zeta \right) \, d\mathcal {L}^{m+n-1} =- \int _\mathcal {O} D^\phi _j \phi \, \zeta \, d \mathcal {L}^{m+n-1}\end{equation*}for every \zeta \in 1_c (\mathcal {O} , \mathbb {R}) and j=2,\dots , m.\end{prop}}{\begin{xmlelement*}{proof}By standard considerations there is a sequence (\phi _h)_{h\in \mathbb {N}} \subset \infty _c (\mathcal {O} , \mathbb {R}) converging uniformly to \phi  on each \mathcal {O} ^{\prime } \Subset \mathcal {O}. We would like to prove that the sequence (D_2^{\phi _h} \phi _h, \dots , D_m^{\phi _h} \phi _h)_{h\in \mathbb {N}} converges to (D^\phi _2 \phi , \dots , D^\phi _m\phi ) in the sense of distributions on each \mathcal {O} ^{\prime } \Subset \mathcal {O}. Indeed, we start to show that\begin{equation}\int _\mathcal {O} \sum _{j=2}^{m} D_j^{\phi } \phi \, \eta _j \, d \mathcal {L}^{m+n-1} =\lim _{h\rightarrow \infty } \int _\mathcal {O} \sum _{j=2}^{m} D_j ^{\phi _h} \phi _h \, \eta _j \, d \mathcal {L}^{m+n-1} \qquad \forall \, \eta \in 1_c (\mathcal {O} , \mathbb {R}^{m-1})\end{equation}We denote by \Phi _h :\mathcal {O} \rightarrow \mathbb {G} the graph map of \hat{\phi }_h=(\phi _h,0,\dots ,0) defined as \Phi _h(a):=i(a)\hat{\phi }_h(i(a)) and \mathcal {E}_h the intrinsic subgraph of \hat{\phi }_h. Therefore, by Proposition \ref {prop4.6} we know that for each \xi =(\xi _1, \dots ,\xi _m)\in 1_c (\hat{ \mathcal {O}} \cdot  \mathbb {R}^m)\begin{equation*}\int _{\mathcal {E}} \mbox{div}_\mathbb {G}\xi \, d \mathcal {L}^{m+n} = \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \xi _1 \circ \Phi - \sum _{j=2}^{m} D_j^{\phi } \phi ( \xi \circ \Phi )_j \, d \mathcal {L}^{m+n-1}\end{equation*}where C(\mathbb {W},>0 is given by Lemma \ref {coro4.3}.Moreover by the uniform convergence of \phi _h to \phi  we conclude that\begin{equation*}\int _{\mathcal {E}} \mbox{div}_\mathbb {G}\xi \, d \mathcal {L}^{m+n} = \lim _{h\rightarrow \infty }\int _{\mathcal {E}_h} \mbox{div}_\mathbb {G}\xi \, d \mathcal {L}^{m+n}\end{equation*}Now we recall that 1 functions are uniformly intrinsic differentiable maps (see Theorem \ref {propC1implicauid}) and consequently they are also locally intrinsic Lipschitz maps (see Proposition \ref {lip1512DDD}). Hence we can apply Proposition \ref {prop4.6} for every \phi _h and we obtain that\begin{equation*}\int _{\mathcal {E}_h} \mbox{div} _\mathbb {G}\xi \, d \mathcal {L}^{m+n} = \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \xi _1 \circ \Phi _h-\sum _{j=2}^{m} D_j^{\phi _h} \phi _h ( \xi \circ \Phi _h)_j \, d \mathcal {L}^{m+n-1}\end{equation*}for every \xi \in 1_c (\hat{ \mathcal {O}} \cdot  \mathbb {R}^m). Finally putting together the last three equalities we have that\begin{equation*}\frac{1}{C(\mathbb {W},} \int _\mathcal {O} \xi _1 \circ \Phi -\sum _{j=2}^{m} D_j^{\phi } \phi ( \xi \circ \Phi )_j \, d \mathcal {L}^{m+n-1}= \lim _{h\rightarrow \infty } \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \xi _1 \circ \Phi _h-\sum _{j=2}^{m} D_j^{\phi _h} \phi _h ( \xi \circ \Phi _h)_j \, d \mathcal {L}^{m+n-1}\end{equation*}for all \xi \in 1_c (\hat{ \mathcal {O}} \cdot  \mathbb {R}^m).\end{xmlelement*}Clearly if we choose \xi _1=0, then\begin{equation}\int _\mathcal {O} \sum _{j=2}^{m} D_j^{\phi } \phi ( \xi \circ \Phi )_j \, d \mathcal {L}^{m+n-1}= \lim _{h\rightarrow \infty } \int _\mathcal {O} \sum _{j=2}^{m} D_j^{\phi _h} \phi _h ( \xi \circ \Phi _h)_j \, d \mathcal {L}^{m+n-1}.\end{equation}If we consider \xi _j(i(a)\cdot (t,0,\dots , 0)):=\eta _j(a)\rho (t) with \eta =(\eta _2, \dots , \eta _m) \in 1_c (\mathcal {O} ,\mathbb {R}^{m-1}) and \rho \in 1_c (\mathbb {R}) such that \rho (t)=1 for all t\in [-\Vert  \phi \Vert _{\mathcal {L}^\infty (\mbox{spt}(\eta ) )}-1,\Vert  \phi \Vert _{\mathcal {L}^\infty (\mbox{spt}(\eta ) )}+1], then \xi =(0, \xi _2,\dots , \xi _m) \in 1_c (\hat{ \mathcal {O}} \cdot  \mathbb {R}^m) and by (\ref {eqADHOC}) we deduce that\begin{equation*}\int _\mathcal {O} \sum _{j=2}^{m} D_j^{\phi } \phi (a) \eta _j(a)\rho (\phi (a)) \, d \mathcal {L}^{m+n-1}(a)= \lim _{h\rightarrow \infty } \int _\mathcal {O} \sum _{j=2}^{m} D_j^{\phi _h} \phi _h (a) \eta _j(a)\rho (\phi _h(a)) \, d \mathcal {L}^{m+n-1}(a).\end{equation*}Hence since \phi _h converges uniformly to \phi , there is \bar{h} \in \mathbb {N} such that for all h\ge \bar{h} and for all a\in  spt(\eta ) we have\begin{enumerate}\item \phi _h (a)\in [-M-1 ,M+1] for M:=\Vert  \phi \Vert _{\mathcal {L}^\infty (\mbox{spt}(\eta ) , \mathbb {R}),}\item \rho (\phi _h(a)) =1;\end{enumerate}and consequently (\ref {72cit}) follows.}Now using again the uniformly convergence of \phi _h to \phi  and (\ref {72cit}) we conclude that for all \eta =(\eta _2,\dots , \eta _m) \in 1_c (\mathcal {O} , \mathbb {R}^{m-1}) and for all j=2,\dots , m\begin{equation*}\begin{aligned}\int _\mathcal {O} D^\phi _j \phi \, \eta _j\, d \mathcal {L}^{m+n-1} & = \lim _{h\rightarrow \infty } \int _\mathcal {O} D^{\phi _h}_j \phi _h\, \eta _j \, d \mathcal {L}^{m+n-1} \\& = - \lim _{h\rightarrow \infty } \int _\mathcal {O} \phi _h \left( \,X_j \eta _j +\phi _h \sum _{s=1}^n b_ {j1 }^{(s)} Y_s \eta _j\right) \, d \mathcal {L}^{m+n-1}\\& = - \int _\mathcal {O} \phi \left( \,X_j \eta _j +\phi \sum _{s=1}^n b_ {j1 }^{(s)} Y_s \eta _j\right) \, d \mathcal {L}^{m+n-1}\\\end{aligned}\end{equation*}Then the thesis follows with \zeta =\eta _j \in 1_c (\mathcal {O}, \mathbb {R}) for each j=2,\dots , m.}\end{equation*}\end{equation}\)
Theorem 5.4 
Let \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 and \(, \) W\( the complementary subgroups defined in (\ref {5.2.0}). Let \) : O  be a locally intrinsic Lipschitz function, where \(\hat{ \mathcal {O}} \subset \mathbb {W}\)  is an open set and \(\phi :\mathcal {O} \rightarrow \mathbb {R}\)  is the map associated to \(\hat{\phi }\)  as in \((\ref {phipsi})\) .
Then there exist \(c=c(\mathbb {G})>0\)  and \(C(\mathbb {W},>0\)  such that the following area formula holds
\(|\partial \mathcal {E}|_\mathbb {G}(\hat{ \mathcal {O}} \cdot = c \, \mathcal {S}^{\mathfrak {q} -1} (\Phi (\mathcal {O} )) =\frac{1}{C(\mathbb {W},} \int _\mathcal {O} \sqrt{1+|\nabla ^{\hat{\phi }} \hat{\phi }|_{\mathbb {R}^{m-1}}^2}\, d \mathcal {L}^{m+n-1}\) 

Thanks to Proposition REF  and Theorem REF  we know that \(|\partial \mathcal {E}|_\mathbb {G}\)  is a Radon measure. Then a classical approximation result ensures the existence of sequence \((\xi _h)_{h\in \mathbb {N}}= (\xi _{h,1},\dots , \xi _{h,m})_{h\in \mathbb {N}}\subset 1_c (\hat{ \mathcal {O}} \cdot  \mathbb {R}^m)\)  with \(|\xi _h|_{\mathbb {R}^{m}}\le 1\)  such that
\(\xi _h \rightarrow \nu _\mathcal {E}\quad |\partial \mathcal {E}|_\mathbb {G}\mbox{ -a.e. in } \hat{ \mathcal {O}} \cdot {equation}and by Lemma \ref {coro4.2} and Lemma \ref {coro4.3} we have that \xi _h \circ \Phi \rightarrow \nu _\mathcal {E}\circ \Phi  a.e. in  \mathcal {O} .\) 
Moreover using () in Proposition REF  we get that for all \(h\in \mathbb {N}\)
\(- \int _{\hat{ \mathcal {O}} \cdot  \langle \xi _h , \nu _\mathcal {E}\rangle \, d|\partial \mathcal {E}|_\mathbb {G}= \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \xi _{h,1 } \circ \Phi -\sum _{j=2}^{m} D_j^{\phi } \phi ( \xi _h \circ \Phi )_j \, d \mathcal {L}^{m+n-1}}Hence taking the limit as h\rightarrow \infty  in the last equality and by (\ref {convanu}) and Lemma \ref {coro4.2} we conclude\begin{equation*}|\partial \mathcal {E}|_\mathbb {G}(\hat{ \mathcal {O}} \cdot = \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \sqrt{1+| \nabla ^{\hat{\phi }} \hat{\phi }|_{\mathbb {R}^{m-1}}^2}\, d \mathcal {L}^{m+n-1}.\end{equation*}Finally, |\partial \mathcal {E}|_\mathbb {G}(\hat{ \mathcal {O}} \cdot = c \, \mathcal {S}^{\mathfrak {q} -1} (\Phi (\mathcal {O} )) is a direct consequence of results of Theorem \ref {Theorem 4.18fssc} and Theorem \ref {teo433}.\) 
Smooth approximation for intrinsic lipschitz maps
In this section we characterize intrinsic Lipschitz functions as maps which can be approximated by a sequence of smooth maps, with pointwise convergent intrinsic gradient.
The classical approximation by convolution does not apply because of the nonlinearity of \(\nabla ^{\hat{\phi }} \hat{\phi }\) . Here we use the similar technique in [2]} in the context of Heisenberg groups. The strategy is somehow indirect, indeed we approximate in Carnot groups of step 2 the intrinsic subgraph of intrinsic Lipschitz function \(\hat{\phi }\) , rather than the function itself. The key point is that the intrinsic subgraph of \(\hat{\phi }\)  is a set of locally finite \(\mathbb {G}\) -perimeter (see Proposition REF ). Precisely, we prove the following Theorem
Theorem 6.1 
Let \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 and \(, \) W\( the complementary subgroups defined in (\ref {5.2.0}). Let \) : O  be a locally intrinsic Lipschitz function, where \(\hat{ \mathcal {O}}\)  is an open subset of \(\mathbb {W}\)  and \(\phi :\mathcal {O} \rightarrow \mathbb {R}\)  is the map associated to \(\hat{\phi }\)  as in \((\ref {phipsi})\) .
Then if \(\nabla ^{\hat{\phi }} \hat{\phi }\in \mathcal {L}^\infty _{loc}(\hat{ \mathcal {O}} ,\mathbb {R}^{m-1})\)  there exists a sequence \((\hat{\phi }_h)_h \subset \infty (\hat{ \mathcal {O}} ,\)  such that for all \(\hat{ \mathcal {O}^{\prime }} \Subset \hat{ \mathcal {O}}\)  there is \(C=C(\mathcal {O}^{\prime })>0\)  such that

\(\hat{\phi }_h\)  uniformly converges to \(\hat{\phi }\)  in \(\hat{ \mathcal {O}^{\prime }} \)

\(|\nabla ^{\hat{\phi }_h} \hat{\phi }_h(a) | \le C\)  \(\forall a\in \hat{ \mathcal {O}^{\prime }} \) , \(h\in \mathbb {N}\)

\(\nabla ^{\hat{\phi }_h} \hat{\phi }_h (a)\rightarrow \nabla ^{\hat{\phi }} \hat{\phi }(a)\)  a.e. \(a\in \hat{ \mathcal {O}^{\prime }}\)


Before stating the approximation theorem we need to recall the following result.
Theorem 6.2 ([3]}) 
Let \(f:\mathbb {R}^{m+n} \rightarrow \mathbb {R}\)  be a strictly convex function and let \((g_h)_h\)  and \(g\)  be in \(\mathcal {L}^1(\Omega , \mathbb {R}^{m+n})\)  where \(\Omega \subset \mathbb {R}^{m+n}\) . If

\(g_h \rightarrow g \, \, \)  weakly in \(\mathcal {L}^1(\Omega , \mathbb {R}^{m+n})\)

\(\int _\Omega f \circ g_h \)  d\( \mathcal {L}^{m+n} \rightarrow \int _\Omega f \circ g \)  d\( \mathcal {L}^{m+n}\)

then \(g_h \rightarrow g \,\)  strongly in \(\mathcal {L}^1(\Omega , \mathbb {R}^{m+n})\) .

Let \(\mathcal {O}^{\prime } \Subset \mathcal {O}\)  and \(M:=\Vert  \phi \Vert _{\mathcal {L}^\infty (\mathcal {O}^{\prime }, \mathbb {R})} <+\infty \) .
We split the proof in several steps. First we use Friedrichs' mollifier to regularize \(\chi _{E }\)  with a family of functions \((f_\alpha )_{\alpha >0}\)  which satisfies Theorem REF . Then we consider the level set of \((f_\alpha )_{\alpha >0}\)  which defines a family of functions \((\phi _\alpha )_{\alpha >0}\) . Finally we prove that \((\phi _\alpha )_{\alpha >0}\)  is the family of functions, up to subsequence, that we were looking for.
\(\mathbf {Step \, 1.}\)  Let \(\mathcal {E}:=\mathcal {E}_{\hat{\phi }}\)  the intrinsic subgraph of \(\hat{\phi }|_{\mathcal {O}^{\prime } }\)  defined as (). For every \(\alpha >0\)  we consider \(f_\alpha :\mathbb {G}\rightarrow \mathbb {R}\)  given by
\(\begin{aligned}f_\alpha (p):= (\rho _\alpha \ast \chi _{\mathcal {E}})(p)& =\int _\mathbb {G}\rho _\alpha (pq^{-1}) \chi _{\mathcal {E}}(q)\, d \mathcal {L}^{m+n}(q)\\& =\int _\mathbb {G}\rho _\alpha (q) \chi _{\mathcal {E}}(q^{-1} p)\, d \mathcal {L}^{m+n}(q)\end{aligned}\) 
where \(\rho _\alpha (p):=\alpha ^{m+2n}\rho (\delta _{1/\alpha }(p))\)  and \(\rho \in \infty _c (\mathcal {U}(0,1), \mathbb {R})\)  is a smooth mollifier with \(\rho (p^{-1})=\rho (-p)=\rho (p)\)  for every \(p\in \mathbb {G}\) .
By properties of convolution in \(\mathbb {G}\)  introduced in [4]}, we know that \(f_\alpha \in \infty _c (\mathbb {G}, \mathbb {R})\)  and spt\((f_\alpha )\subset \mathcal {U}(0,\alpha )\cdot \)  spt\((\chi _{\mathcal {E}})\)  for every \(\alpha >0\) . Precisely, for all \(\alpha >0\)
\(f_\alpha (p)\in [0,1] \quad \mbox{ for all } p\in \mathbb {G}\) 
and for all sufficiently small \(\alpha >0\)
\(f_\alpha (p)=1 \quad \mbox{ for all } p=(p_1,\dots , p_{m+n})\in \mathbb {G}\, \mbox{ such that } p_1 \le -2M\) 
Moreover by definition of \(\mathcal {E}\)  we know that \(\mathcal {E}\)  is an open set in \(\mathbb {G}\)  and
\(\mbox{ spt} (\chi _{\mathcal {E}}) = \mbox{clos} ( \mathcal {E}) \subset \lbrace a\cdot (t,0,\dots , 0 )\in \mbox{clos} (\hat{ \mathcal {O}^{\prime }} ) \cdot :\, t \le M \rbrace .\) 
As a consequence
\(\mbox{ spt} (f_\alpha ) \subset \overline{\mathcal {U}(0,\alpha )}\cdot \mbox{ spt} (\chi _{\mathcal {E}})\subset \lbrace a\cdot (t,0,\dots , 0 )\in \hat{ \mathcal {O}^{\prime }} \cdot :\, t < 2M \rbrace  \quad \mbox{ for } \alpha <M\) 
and
\(f_\alpha (p)=0 \quad \mbox{ for all } p=(p_1,\dots , p_{m+n})\in \mathbb {G}\, \mbox{ such that } p_1 \ge 2M\) 
i.e. \(f_\alpha \)  is constant far from the graph of \(\hat{\phi }\) , so \((\ref {conv})\)  is extended only in a neighborhood of the graphs itself.
\(\mathbf {Step \, 2.}\)  For every \(\alpha >0\)  and \(c\in (0,1)\) , let
\(S_\alpha := \left\lbrace  p=(p_1,\dots , p_{m+n})\in \hat{ \mathcal {O}^{\prime }} \cdot  : \, p_1 \in (-2M,2M),\, f_\alpha (p)=c \right\rbrace .\) 
We note that if
\(\mbox{rank} \nabla _\mathbb {G}f_\alpha (p)=1, \quad \mbox{for all } p\in S_\alpha \) 
then \(S_\alpha \)  is the level set of a map \(f_\alpha \in \infty (\mathbb {G}, \mathbb {R})\)  and so \(f_\alpha \in 1_\mathbb {G}(\mathbb {G}, \mathbb {R})\)  such that \(\mbox{rank} \nabla _\mathbb {G}f_\alpha \)  is maximum, i.e. \(S_\alpha \)  is \(\mathbb {G}\) -regular hypersurface. More precisely, we show that
\(X_1f_\alpha (p) <0 \quad \mbox{for all } p\in S_\alpha \) 
where \(X_1f_\alpha (p)\)  is the first component of horizontal gradient \(\nabla _\mathbb {G}f_\alpha \)  of \( f_\alpha \)  in \(p\) . Indeed let \(\mathcal {A} ^{\prime }:=\lbrace p= (p_1,\dots , p_{m+n})\in \hat{ \mathcal {O}^{\prime }} \cdot  :\, p_1\in (-3M,3M) \rbrace \)  and \(\varphi \in \infty _c(\mathcal {A} ^{\prime } , \mathbb {R})\) , then
\(\begin{aligned}\langle X_1f_\alpha , \varphi \rangle &=-\int _{ \mathcal {A} ^{\prime } } f_\alpha (q)X_1 \varphi (q) \, d\mathcal {L}^{m+n} (q)\\&=-\int _{ \overline{\mathcal {U}(0,\alpha )} } \rho _\alpha (p) \, d\mathcal {L}^{m+n} (p) \int _{ \mathcal {A} ^{\prime } } \chi _\mathcal {E}(p^{-1}q) X_1 \varphi (q)\, d\mathcal {L}^{m+n} (q)\\&=-\int _{ \overline{\mathcal {U}(0,\alpha )} } \rho _\alpha (p) \, d\mathcal {L}^{m+n} (p) \int _{\tau _{P^{-1}} (\mathcal {A} ^{\prime }) } \chi _\mathcal {E}(q) X_1 \varphi (pq)\, d\mathcal {L}^{m+n} (q)\\\end{aligned}\) 
With the notation \(\varphi _p(q):=\varphi (pq)\)  we have \(X_1(\varphi (pq))= X_1 \varphi _p (q)\)  because \(X_1\)  is left-invariant; moreover \(\varphi _p \in \infty _c (\tau _{p^{-1}} \mathcal {A} ^{\prime } , \mathbb {R})\)  and by integration by parts, we obtain
\(\begin{aligned}\int _{\tau _{P^{-1}} (\mathcal {A} ^{\prime }) } \chi _\mathcal {E}(q) X_1 \varphi _P(q)\, d\mathcal {L}^{m+n} (q) = - \int _{\tau _{p^{-1}} (\mathcal {A} ^{\prime }) } \nu ^{(1)}_{\mathcal {E}} (q) \varphi _P(q)\, d |\partial \mathcal {E}|_\mathbb {G}(q)\end{aligned}\) 
Since \(\mbox{spt}(\varphi _P) \Subset \tau _{P^{-1}} (\mathcal {A} ^{\prime })\)  and \(P\in \overline{\mathcal {U}(0,\alpha )}\)  if \(\alpha \)  is small enough, we can replace \(\tau _{p^{-1}} (\mathcal {A} ^{\prime })\)  by \(\mathcal {A} ^{\prime }\) . Thus by Fubini-Tonelli Theorem and a change of variable, we get
\(\begin{aligned}\langle X_1f_\alpha , \varphi \rangle &=\int _{\mathcal {A} ^{\prime } } \nu ^{(1)}_{\mathcal {E}} (q) \, d |\partial \mathcal {E}|_\mathbb {G}(q) \left( \int _{\mathbb {G}} \rho _\alpha (p) \varphi (pq) \, d\mathcal {L}^{m+n} (p) \right)\\\end{aligned}\) 
Then for all \(p\in \mathcal {A} ^{\prime }\)  and for all \(\alpha >0\)  small enough
\(\begin{aligned}X_1f_\alpha (p) & = \int _{ \mathcal {A} ^{\prime } }\rho _\alpha (p q^{-1}) \nu ^{(1)}_{\mathcal {E}} (q)\, d |\partial \mathcal {E}|_\mathbb {G}(q)\\&= \int _{\mathcal {U}(0,\alpha )\cdot P }\rho _\alpha (p q^{-1}) \nu ^{(1)}_{\mathcal {E}} (q)\, d |\partial \mathcal {E}|_\mathbb {G}(q)\end{aligned}\) 
In particular we immediately deduce from \((\ref {83cit})\)  the following assertion: For all couple \((\hat{ \mathcal {O}^{\prime }} ,\hat{ \mathcal {O}} _0^{\prime })\)  of open and bounded subsets of \(\mathbb {W}\)  with \(\hat{ \mathcal {O}^{\prime }} \Subset \hat{ \mathcal {O}} _0^{\prime }\)  there is \(\bar{\alpha }=\alpha (\hat{ \mathcal {O}} _0^{\prime })>0\)  such that for every \(0<\alpha <\bar{\alpha }\)
\(\int _{ \mathcal {A} }|\nabla _\mathbb {G}f_\alpha |\, d \mathcal {L}^{m+n} \le |\partial \mathcal {E}|_\mathbb {G}( \mathcal {A}_0 ).\) 
where \( \mathcal {A}:=\lbrace p= (p_1,\dots , p_{m+n})\in \hat{ \mathcal {O}^{\prime }} \cdot  :\, p_1\in (-2M,2M) \rbrace \)  and \( \mathcal {A}_0 :=\lbrace p= (p_1,\dots , p_{m+n})\in \hat{ \mathcal {O}}_0^{\prime } \cdot  :\, p_1\in (-2M,2M) \rbrace \) . Moreover since Lemma \(\ref {coro4.2}\)  we get
\(\nu ^{(1)}_{\mathcal {E}} \circ \Phi =-\frac{1}{\sqrt{1+|\nabla ^{\hat{\phi }} \hat{\phi }|_{ \mathbb {R}^{m-1}}^2}}\quad \mbox{in } \mathcal {O}^{\prime }\) 
and if we put
\(I_\alpha (p):= \int _{ \mathcal {U}(0,\alpha )\cdot p}\rho _\alpha (pq^{-1})\, d |\partial \mathcal {E}|_\mathbb {G}(q)\) 
then using \((\ref {83cit})\)  we have
\(X_1f_\alpha (p) \le -\frac{1}{\sqrt{1+\Vert \nabla ^{\hat{\phi }} \hat{\phi }\Vert ^2_{\mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }} , \mathbb {R}^{m-1})}}}I_\alpha (p) \quad \mbox{ for all } p \in \mathcal {A}^{\prime } .\) 
Moreover for every fixed \(c\in (0,1)\)  and for all \(p\in \mathcal {A}\)  such that \(f_\alpha (p)=c\) , then
\(\mathcal {L}^{m+n}((\mathcal {U}(0,\alpha )\cdot p) \cap \mathcal {E})>0 \quad \mbox{and} \quad \mathcal {L}^{m+n}((\mathcal {U}(0,\alpha )\cdot p) \cap \mathcal {E}^c )>0.\) 
Indeed, by contradiction, if we assume \( \mathcal {L}^{m+n}((\mathcal {U}(0,\alpha )\cdot p) \cap \mathcal {E})=0\) . Then because \(\mathcal {E}\)  is open, we can assume \((\mathcal {U}(0,\alpha )\cdot p) \cap \mathcal {E}= \emptyset \) . Hence by the definition of convolution, we have that \(f_\alpha (p)=0 \ne c\)  and then a contradiction. In the similar way it follows that \(f_\alpha (p)=1\)  if we suppose by contradiction \( \mathcal {L}^{m+n}((\mathcal {U}(0,\alpha )\cdot p) \cap \mathcal {E}^c)=0\) .
Now by \((\ref {cit88})\)  and by Theorem REF , we have that \(|\partial \mathcal {E}|_\mathbb {G}((\mathcal {U}(0,\alpha )\cdot p )>0\)  and \(I_\alpha (p )>0 \)  for \(p\in \mathcal {A} \)  with \(f_\alpha (p)=c\) . As a consequence using \((\ref {cit87})\) , \((\ref {cit85})\)  holds and \(S_\alpha \)  is a \(\mathbb {G}\) -regular hypersurface.
\(\mathbf {Step \, 3.}\)  Now we show that \((S_\alpha )_\alpha \)  implicitly defines a sequence \((\hat{\phi }_\alpha )_{\alpha }\)  with \(\hat{\phi }_\alpha =(\phi _\alpha ,0,\dots , 0)\)  and ,up to subsequence, it is the family of functions which we were looking for.
Fix \(\alpha \)  and \(c\in (0,1)\) . Because \(f_\alpha \)  is continuous map such that
\(\begin{aligned}f_\alpha ( i(a) \cdot (-2M,0,\dots , 0)) & =1>c \\f_\alpha (i(a)\cdot (2M,0,\dots , 0)) & = 0<c\end{aligned}\) 
for all \(a\in \mathcal {O}^{\prime } \) , there is \(t_a\in (-2M,2M)\)  such that \(f_\alpha ( i(a) \cdot (t_a,0 , \dots , 0))=c\) . In particular since
\(X_1f_\alpha ( i(a) \cdot (t,0,\dots , 0) )\le 0 \quad \forall \, t<t_a \, \qquad \mbox{ and } \qquad X_1f_\alpha ( i(a) \cdot (t_a,0,\dots , 0)) < 0\) 
we have that \(t_a\)  is unique and \( \lbrace  t\in (-2M,2M) \, :\, f_\alpha ( i(a) \cdot (t,0,\dots , 0))>c \rbrace =(-2M,t_a)\) .
In the similar way in Lemma 4.8 in [5]}, we define \(\hat{\phi }_\alpha (i(a)):=(\phi _\alpha (a),0,\dots , 0)\)  with \(\phi _\alpha (a):=t_a\)  and
\(\mathcal {E}_\alpha =\mathcal {E}_{\alpha , c}:= \lbrace  (i(a) \cdot (t,0,\dots , 0)) \in \hat{ \mathcal {O}} \cdot  :\, f_\alpha (i(a)\cdot (t,0,\dots , 0))>c\rbrace .\) 
Then \(\phi _\alpha :\mathcal {O}^{\prime } \rightarrow [-2M, 2M]\)  and
\(\mathcal {E}_\alpha \cap \mathcal {A}^{\prime \prime } =\mathcal {E}_{\hat{\phi }_\alpha } \cap \mathcal {A} ^{\prime \prime },\) 
where \( \mathcal {A}^{\prime \prime } :=\lbrace p\in \hat{ \mathcal {O}^{\prime }} \cdot  :\, p_1\in [-2M, 2M] \rbrace \) . Moreover recalling that \(c\in (0,1)\) , from \((\ref {cit76})\)  and \((\ref {cit79})\)  we get
\(S_\alpha = \mathrm {graph}\,(\hat{\phi }_\alpha ) =\partial \mathcal {E}_\alpha \cap ( \hat{ \mathcal {O}^{\prime }} \cdot \) 
and by Theorem REF  \(\hat{\phi }_\alpha \)  is uniformly intrinsic differentiable in \(\hat{ \mathcal {O}^{\prime }} \) .
Let \(\tilde{f} _\alpha :\mathbb {R}^{m+n} \rightarrow \mathbb {R}\)  defined by
\( \tilde{f}_\alpha (t,a ):=f_\alpha (i(a)\cdot (t,0,\dots , 0)).\) 
By \(f_\alpha \in \infty (\mathbb {G}, \mathbb {R})\)  and (REF ), \( \tilde{f}_\alpha \)  is \(\infty \)  map such that \(\frac{ \partial \tilde{f}_\alpha }{\partial p_1}(p) = X_1 f_\alpha (p) \ne 0\)  for all \(p\in S_\alpha \) . Hence we apply the classical Implicit Function Theorem to \(\tilde{f} _\alpha \)  and we conclude that also \(\hat{\phi }_\alpha \)  is \(\infty \) .
\(\mathbf {Step \, 4.}\)  Firstly we prove that \((\hat{\phi }_\alpha )_\alpha \) , up to subsequence, converges uniformly on \(\hat{ \mathcal {O}^{\prime }}\) , i.e. the condition (1) of the thesis follows.
In particular, we would like to show that there is a constant \(L>0\)  dependent on \( \Vert \nabla ^{\hat{\phi }} \hat{\phi }\Vert _{\mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }} , \mathbb {R}^{m-1})} \) , \(\mathcal {B}^{(1)}, \dots ,\mathcal {B}^{(n)}\)  such that for all \(\alpha >0\)  sufficiently small and all \(A\in \mathcal {O}^{\prime }\) , it holds
\(|\phi _\alpha (a)-\phi (a)|\le L\alpha .\) 
Therefore \((\phi _\alpha )_\alpha \) , up to subsequence, converges uniformly on \(\mathcal {O}^{\prime }\) . This is a direct consequence of Theorem REF . Here \(\nu ^\perp (p) =\mathbf {P}_\mathbb {W}(p)\) , \(\nu (p) =\mathbf {P}_p)\)  and \(E:=\mathcal {E}_{\hat{\phi }_\alpha } \)  is the intrinsic subgraph of \(\hat{\phi }_\alpha \) . Because \(\hat{\phi }_\alpha \)  is locally intrinsic Lipschitz, \(\mathcal {E}_{\hat{\phi }_\alpha } \)  is also a finite \(\mathbb {G}\) -perimeter set (see Theorem REF ).
For \(\beta \)  fixed as in Theorem REF  there is \(L=L(\beta , \mathcal {B}_M)>0\) , where \(\mathcal {B}_{M} = \max \lbrace  b_{ij}^{(s)} \, | \, i,j=1,\dots , m \, , s=1,\dots , n \rbrace \)  such that
\(\mathcal {U}((-L,0,\dots ,0),1) \subset \lbrace  q\, :\, \Vert  \mathbf {P}_\mathbb {W}q\Vert <-\beta \mathbf {P}_q̭ \rbrace  \quad \mbox{and} \quad \mathcal {U}((L,0,\dots ,0),1) \subset \lbrace  q\, :\, \Vert  \mathbf {P}_\mathbb {W}q\Vert <\beta \mathbf {P}_q̭ \rbrace \) 
Let \(p_a:= i(a) \cdot \hat{\phi }(i(a))\) , \(p^{\prime }_a:= i(a) \cdot (\phi (a)-\alpha L ,0,\dots , 0)\)  and \(p^{\prime \prime }_a:= i(a) \cdot (\phi (a)+\alpha L ,0,\dots , 0)\)  with \(a \in \mathcal {O}^{\prime } \)  and \(\alpha \in (0,1]\) . Notice that if we put \(r_0:= M+L+\max _{(x,y)\in \mathcal {O}^{\prime } }|x|_{\mathbb {R}^{m-1}}+ \max _{(x,y)\in \mathcal {O}^{\prime } }|y|_{\mathbb {R}^n}^{1/2} +\frac{1}{2} \mathcal {B}_M^{1/2} \max _{(x,y)\in \mathcal {O}^{\prime } }|x|_{\mathbb {R}^{m-1}}^{1/2} (M+L)^{1/2}< +\infty \)  then
\(\Vert  p^{\prime }_a\Vert  \le r_0,\quad \Vert  p^{\prime \prime }_a\Vert  \le r_0,\) 
for all \(a\in \mathcal {O}^{\prime }\) . Moreover by standard considerations (see [6]}) we know that for every \(q\in \mathcal {U}(0,r_0)\)  there exists \(c(r_0)>0\)  such that
\(\mathcal {U}(0,r)\cdot q \subset \mathcal {U}(q,c(r_0)\sqrt{r} ), \quad \forall \, r \in (0,1).\) 
So if we consider \(a\in \mathcal {O}^{\prime }\)  and \(\alpha \in (0,1]\) , we conclude that
\(\begin{aligned}\mathcal {U}(0,\alpha )\cdot p^{\prime }_a & \subset \mathcal {U}(p^{\prime }_a,c(r_0)\sqrt{\alpha }) \subset \lbrace  q\, :\, \Vert  \mathbf {P}_\mathbb {W}(p_a ^{-1} Q)\Vert <-\beta \Vert \mathbf {P}_p_a ^{-1} q) \Vert  \rbrace  \subset \mathcal {E}_{\hat{\phi }_\alpha }\\\mathcal {U}(0,\alpha )\cdot p^{\prime \prime }_a& \subset \mathcal {U}(p^{\prime \prime }_a,c(r_0)\sqrt{\alpha }) \subset \lbrace  q\, :\, \Vert  \mathbf {P}_\mathbb {W}(p_a ^{-1} Q)\Vert <\beta \Vert \mathbf {P}_p_a^{-1} q)\Vert  \rbrace  \subset \mathbb {G}- \mathcal {E}_{\hat{\phi }_\alpha }\end{aligned}\) 
More precisely, by definition of \(f_\alpha \)  we have
\(f_\alpha (p^{\prime }_a)=1, \quad \quad f_\alpha \left( i(a) \cdot \hat{\phi }_\alpha (i(a)) \right)=c, \quad \quad f_\alpha (p^{\prime \prime }_a)=0\) 
and so by \((\ref {90cit})\) , \((\ref {91cit})\)  and \((\ref {cit85})\)  we deduce
\(\phi (a)-\alpha L \le \phi _\alpha (a) \le \phi (a)+\alpha L\quad \mbox{ for all } \alpha \in (0,1], \, a\in \mathcal {O}^{\prime }.\) 
Hence (REF ) holds and consequently the condition (1) of the thesis is true.
\(\mathbf {Step \, 5.}\)  We prove that \((\hat{\phi }_\alpha )_{\alpha }\)  has a subsequence \((\hat{\phi }_k)_k\)  such that \(|\nabla ^{\hat{\phi }_k} \hat{\phi }_k |_{\mathbb {R}^{m-1}} \le \Vert  \nabla ^{\hat{\phi }} \hat{\phi }\Vert _{\mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }} ,\mathbb {R}^{m-1})}\)  for each \(k\in \mathbb {N}\)  on \(\mathcal {O} ^{\prime }\) , i.e. the condition \((2)\)  of the thesis holds.
Let \(\hat{\nabla }_\mathbb {G}f_\alpha :=(X_2f_\alpha , \dots ,X_mf_\alpha )\)  and \(\hat{\nu }_{\mathcal {E}} =(\nu ^{(2)}_{\mathcal {E}},\dots , \nu ^{(m)}_{\mathcal {E}})\) . Arguing as in Step 2 and by (), it follows
\(\begin{aligned}| \nabla ^{\hat{\phi }_\alpha } \hat{\phi }_\alpha (p)|_{\mathbb {R}^{m-1}} &= \frac{|\hat{\nabla }_\mathbb {G}f_\alpha (p)|_{\mathbb {R}^{m-1}}}{|X_1f_\alpha (p)|} \le \frac{1}{|X_1f_\alpha (p)|} \int _{ \mathcal {U}(0,\alpha )\cdot p} |\hat{\nu }_{\mathcal {E}} (q)| |\rho _\alpha (p\cdot q^{-1})| \, d |\partial \mathcal {E}|(q)\\& \le I_\alpha (p) \frac{ \Vert  \nabla ^{\hat{\phi }} \hat{\phi }\Vert _{\mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }},\mathbb {R}^{m-1})} }{|X_1f_\alpha (p)| \sqrt{1+ \Vert  \nabla ^{\hat{\phi }} \hat{\phi }\Vert ^2_{\mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }} ,\mathbb {R}^{m-1})} }} \\\end{aligned}\) 
and consequently by \((\ref {cit87})\)
\(| \nabla ^{\hat{\phi }_\alpha } \hat{\phi }_\alpha (p)|_{\mathbb {R}^{m-1}} \le \Vert  \nabla ^{\hat{\phi }} \hat{\phi }\Vert _{ \mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }} ,\mathbb {R}^{m-1})},\) 
for all \(p\in \hat{ \mathcal {O}^{\prime }}\) , as desired.
\(\mathbf {Step \, 6.}\)  Now we show that \((\hat{\phi }_\alpha )_\alpha \) , up to a subsequence, satisfying the condition \((3)\)  of the thesis, i.e. there exists a sequence \((\alpha _h)_h \subset (0,+\infty )\)  such that let \(\hat{\phi }_h \equiv \hat{\phi }_{\alpha _h}\)  we have
\(\nabla ^{\hat{\phi }_h} \hat{\phi }_h \rightarrow \nabla ^{\hat{\phi }} \hat{\phi }\quad \mbox{ a.e. in } \hat{ \mathcal {O}^{\prime }}.\) 
It is sufficient to show that there is a positive sequence \((\alpha _h)_h\)  converging to 0 such that there exists
\(\lim _{h\rightarrow \infty } \int _{ \mathcal {O}^{\prime }} \sqrt{ 1+|\nabla ^{\hat{\phi }_h} \hat{\phi }_h|_{\mathbb {R}^{m-1}} ^2} \, d \mathcal {L}^{m+n-1} = \int _{ \mathcal {O}^{\prime }} \sqrt{ 1+|\nabla ^{\hat{\phi }} \hat{\phi }|_{\mathbb {R}^{m-1}} ^2} \, d \mathcal {L}^{m+n-1}\) 
In fact, up a subsequence, using \((\ref {cit93})\)  and Proposition \(\ref {theo4.7}\)  we can assume that the sequence in \((\ref {citt98})\)  also satisfies
\(\nabla ^{\hat{\phi }_h} \hat{\phi }_h \rightarrow \nabla ^{\hat{\phi }} \hat{\phi }\quad \mbox{ weakly in }\, \mathcal {L}^1( \hat{ \mathcal {O}^{\prime }}, \mathbb {R}^{m-1} ).\) 
Consequently by Theorem  REF
\(\nabla ^{\hat{\phi }_h} \hat{\phi }_h \rightarrow \nabla ^{\hat{\phi }} \hat{\phi }\quad \mbox{ strongly in }\, \mathcal {L}^1( \hat{ \mathcal {O}^{\prime }}, \mathbb {R}^{m-1} )\) 
and so, up a subsequence, \((\ref {citt97})\)  holds.
In particular in order to show \((\ref {citt98})\)  we need to prove that there exist \(\hat{c} \in (0,1)\)  and a positive sequence \((\alpha _h)_h\)  converging to 0 satisfying
\(\exists \, \lim _{h\rightarrow \infty } |\partial \mathcal {E}_{\alpha _h, \hat{c} }|_\mathbb {G}( \mathcal {A}) = |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A})\) 
where \(\mathcal {A}\)  is defined as in Step 2. Since the semicontinuity of \(\mathbb {G}\) -perimeter measure and Step 3 we obtain
\(|\partial \mathcal {E}|_\mathbb {G}(\mathcal {A} ) \le \liminf _{\alpha \rightarrow 0^+} |\partial \mathcal {E}_{\alpha , c }|_\mathbb {G}(\mathcal {A} )\) 
for every \(c\in (0,1)\) . More precisely by the last inequality and the coarea formula we get
\(\begin{aligned}|\partial \mathcal {E}|_\mathbb {G}(\mathcal {A}) & \le \int _0^1 \liminf _{\alpha \rightarrow 0^+} |\partial \mathcal {E}_{\alpha , c }|_\mathbb {G}(\mathcal {A} )\, dc \le \liminf _{\alpha \rightarrow 0^+} \int _0^1 |\partial \mathcal {E}_{\alpha , c }|_\mathbb {G}(\mathcal {A} )\, dc \\ & \quad = \liminf _{\alpha \rightarrow 0^+} \int _{\mathcal {A} } | \nabla _\mathbb {G}f_\alpha |\, d \mathcal {L}^{m+n} =: I(\hat{ \mathcal {O}^{\prime }} , c ).\end{aligned}\) 
Now using Step 2 we know that for every \(\hat{ \mathcal {O}}_0^{\prime }\)  such that \(\hat{ \mathcal {O}^{\prime }}\Subset \hat{ \mathcal {O}} _0^{\prime } \)  open and bounded there exists \((\alpha _h)_h \subset (0,+ \infty )\)  which converges to 0 and \( \bar{h} (\hat{ \mathcal {O}}_0^{\prime })>0\)  such that for all \(h\le \bar{h} (\hat{ \mathcal {O}}_0^{\prime })\)  we have
\(\int _{\mathcal {A} } | \nabla _\mathbb {G}f_{\alpha _h}| \, d \mathcal {L}^{m+n} \le |\partial \mathcal {E}|_\mathbb {G}( \mathcal {A}_0)\) 
where \(\mathcal {A}, \mathcal {A}_0\)  are defined as in Step 2. Consequently
\(I(\hat{ \mathcal {O}^{\prime }}, c ) \le |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A}_0)\) 
for all \(c\in (0,1)\)  and \(\hat{ \mathcal {O}}_0^{\prime }\)  such that \(\hat{ \mathcal {O}^{\prime }} \Subset \hat{ \mathcal {O}}_0^{\prime } \)  open and bounded. Moreover by \(|\partial \mathcal {E}|_\mathbb {G}\)  is a Radon measure then by standard approximation argument we can rewrite \((\ref {cit109})\)  with \(\hat{ \mathcal {O}^{\prime }}\)  instead of \(\hat{ \mathcal {O}}_0^{\prime }\) . In particular using \((\ref {cit106})\)  we have that a.e. \(c\in (0,1)\)
\(\liminf _{\alpha \rightarrow 0^+} |\partial \mathcal {E}_{\alpha , c }|_\mathbb {G}(\mathcal {A} )= |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A} ).\) 
Hence there exists \(\hat{c} \in (0,1)\)  and a positive sequence \((\alpha _h)_h\)  converging to 0 such that \((\ref {cit101})\)  follows.
Finally because \(\Vert  \phi \Vert _{\mathcal {L}^\infty (\mathcal {O} ^{\prime } , \mathbb {R}) } = M\)  we get
\(\partial \mathcal {E}\cap \mathcal {A} _1 = \partial \mathcal {E}\cap \mathcal {A} _2 = \emptyset \) 
where \( \mathcal {A}_1:=\lbrace p \in \hat{ \mathcal {O}^{\prime }} \cdot  :\, p_1 \le -2M \rbrace \)  and \(\mathcal {A}_2:=\lbrace p\in \hat{ \mathcal {O}^{\prime }} \cdot  :\, p_1 \ge 2M \rbrace \) .
Hence since Proposition \(\ref {citteo16OK}\)  and well-know \(\mathbb {G}\) -perimeter properties it follows that
\(\begin{aligned}\frac{1}{C(\mathbb {W},} \int _{ \mathcal {O}^{\prime }} \sqrt{1+| \nabla ^{\hat{\phi }} \hat{\phi }|_{\mathbb {R}^{m-1}} ^2} \, d\mathcal {L}^{m+n-1} & =|\partial \mathcal {E}|_\mathbb {G}( \hat{ \mathcal {O}^{\prime }} \cdot  \\& = |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A} _1 )+ |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A} )+ |\partial \mathcal {E}| ( \mathcal {A}_2 )\\& = |\partial \mathcal {E}|_\mathbb {G}( \partial \mathcal {E}\cap \mathcal {A}_1) + |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A} ) + |\partial \mathcal {E}|_\mathbb {G}( \partial \mathcal {E}\cap \mathcal {A}_2 )\\&= |\partial \mathcal {E}|_\mathbb {G}( \mathcal {A} ).\end{aligned}\) 
In the same way, using also \((\ref {cit76})\) , \((\ref {cit79})\)  and \((\ref {90cit})\)
\(|\partial \mathcal {E}_{\alpha _h , c} |_\mathbb {G}( \mathcal {A} )= |\partial \mathcal {E}_{\hat{\phi }_h }|_\mathbb {G}( \mathcal {A} )\) 
and
\(|\partial \mathcal {E}_{\hat{\phi }_h }|_\mathbb {G}( \hat{ \mathcal {O}^{\prime }} \cdot = \frac{1}{C(\mathbb {W},} \int _{ \mathcal {O}^{\prime }} \sqrt{1+|\nabla ^{\hat{\phi }_h} \hat{\phi }_h |_{\mathbb {R}^{m-1}} ^2} \, \, d \mathcal {L}^{m+n-1}\) 
with \(\hat{\phi }_h = \hat{\phi }_{\alpha _h }\) . Now by the last three equalities and \((\ref {cit101})\) , \((\ref {citt98})\)  holds. Consequently \((\ref {citt97})\)  is true and the proof of theorem is complete.

Intrinsic Lipschitz Function and Distributional Solution
The main results in this section are Theorem REF  and Theorem REF . We show that a map \(\hat{\phi }=(\phi , 0 \dots , 0): \hat{ \mathcal {O}} \rightarrow  is a locally intrinsic Lipschitz function if and only if \)\( is locally \) 1/2 \(-Hölder continuous and it is also distributional solution of system \)\(\left(D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi \right) =w \quad \mbox{in } \mathcal {O}.\)\(More precisely, we generalize Theorem 1.1 on \cite {biblio17} valid in Heisenberg groups.\)
Theorem 7.1 
Let \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 and \(, \) W\( the complementary subgroups defined in (\ref {5.2.0}). Let \) : O  be a locally intrinsic Lipschitz function, where \(\hat{ \mathcal {O}}\)  is an open subset of \(\mathbb {W}\)  and \(\phi :\mathcal {O} \rightarrow \mathbb {R}\)  is the map associated to \(\hat{\phi }\)  as in \((\ref {phipsi})\) .
Then if \((D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi ) \in \mathcal {L}_{loc}^\infty (\mathcal {O} ,\mathbb {R}^{m-1})\) , we have that \(\phi \)  is a distributional solution of
\((D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi ) =w\)  for a suitable \(w\in \mathcal {L}^\infty _{loc} (\mathcal {O} , \mathbb {R}^{m-1})\)  such that \(w(a)=(D^{\phi }_2\phi (a),\dots , D^{\phi }_m\phi (a) )\)  \(\mathcal {L}^{m+n-1} \) -a.e. in \(\mathcal {O}\) .

By Theorem \(\ref {cmpssteo17}\)  we know that there exist \((\phi _h)_{h\in \mathbb {N}} \subset \infty (\mathcal {O} ,\mathbb {R})\)  such that for all \(\mathcal {O}^{\prime } \Subset \mathcal {O}\)  we have

\( \phi _h\)  uniformly converges to \( \phi \)  in \(\mathcal {O}^{\prime } \)

\(|(D^{ \phi _h}_2 \phi _h(a), \dots , D^{ \phi _h}_m \phi _h(a)) |_{\mathbb {R}^{m-1}} \le \Vert  (D^{ \phi }_2 \phi (a), \dots , D^{ \phi }_m \phi (a)) \Vert _{\mathcal {L}^\infty (\mathcal {O}^{\prime },\mathbb {R}^{m-1})}\)  \(\forall a\in \mathcal {O} ^{\prime }\) , \(h\in \mathbb {N}\)

for each \(j=2,\dots , m\) , we have \(D^{ \phi _h}_j \phi _h (a)\rightarrow D^{ \phi }_j \phi (a)\)  a.e. \(a\in \mathcal {O}^{\prime }\)

Let \(w_h =(w_{2,h}, \dots , w_{m,h}):=(D_2^{\phi _h} \phi _h, \dots ,D_m^{\phi _h} \phi _h)\) . Observe that for each \(h\in \mathbb {N}\)  and \(\zeta \in \infty _c (\mathcal {O} ,\mathbb {R})\)
\(\int _\mathcal {O} \phi _h \left( \,X_j \zeta +\phi _h \sum _{s=1}^n b_ {j1 }^{(s)} Y_s \zeta \right) \, d\mathcal {L}^{m+n-1} = - \int _\mathcal {O} w_{j,h} \zeta \, d \mathcal {L}^{m+n-1}\) 
for all \(j=2,\dots , m\) . Getting to the limit for \(h\rightarrow \infty \)  we have
\(\int _\mathcal {O} \phi \left( \,X_j \zeta +\phi \sum _{s=1}^n b_ {j1 }^{(s)} Y_s \zeta \right) \, d\mathcal {L}^{m+n-1} = - \int _\mathcal {O} w_{j} \zeta \, d \mathcal {L}^{m+n-1}.\) 
Hence \(\phi \)  is a distributional solution of \((D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi ) =w\)  in \(\mathcal {O}\) .
Theorem 7.2 
Let \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 and \(, \) W\( the complementary subgroups defined in (\ref {5.2.0}). Let \) :O  be a continuous map where \(\hat{\mathcal {O}} \)  is open in \(\mathbb {W}\)  and \(\phi :\mathcal {O} \rightarrow \mathbb {R}\)  is the map associated to \(\hat{\phi }\)  as in \((\ref {phipsi})\) . We also assume that

\(\phi \)  is a continuous distributional solution of \((D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi ) =w\)  in \(\mathcal {O} \)  with \(w\in \mathcal {L}^\infty _{loc}(\mathcal {O} ,\mathbb {R}^{m-1})\)

\(\phi \)  is locally \(1/2 \) -Hölder continuous along the vertical components with Hölder's constant \(C_h>0\)

Then \(\hat{\phi }\)  is locally intrinsic Lipschitz function.

The proof of Theorem REF  relies on a preliminary result about the distributional solutions of the problem \((D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi ) =w\)  in \(\mathcal {O} \) .
It is convenient to introduce the following notation: for \(j=2,\dots ,m \)  and
\(\hat{x}_j=(\hat{x}_2, \dots , \hat{x}_{j-1} ,\hat{x}_{j+1}, \dots , \hat{x}_{m}) \in \mathbb {R}^{m-2}\)  fixed, we denote by \((t,\hat{x}_j) :=(\hat{x}_2, \dots , \hat{x}_{j-1}, t ,\hat{x}_{j+1}, \dots , \hat{x}_{m} )\)
Lemma 7.3 
Under the same assumptions of Theorem \(\ref {lemma5.4bcsc}\) , for all \(j=2,\dots , m\)  there is \(C=C(\Vert  w_j\Vert _{\mathcal {L}^\infty (\mathcal {O}, \mathbb {R})}, C_h, b_ {j1 }^{(1)},\dots , b_ {j1 }^{(n)})>0\)  such that \(\phi \)  is a C-Lipschitz map along any characteristic line \(\gamma _{j}=(\gamma _{j1}, \dots , \gamma _{jn}):[-T, T ]\rightarrow \mathbb {R}^n\)  satisfying
\(\dot{\gamma }_{js} (t)= b_ {j1 }^{(s)} \phi (t,\hat{x}_j, \gamma _j(t))+ \frac{1}{2}\sum _{l=2}^{m}b^{(s)}_{jl}(\hat{x}_j )_l \quad \mbox{ for all } s=1,\dots , n\) 
with \(t\in [-T , T ]\)  and \(\hat{x}_j\in \mathbb {R}^{m-2}\)  fixed.

Fix \(j=2,\dots ,m\)  and \(\hat{x}_j \in \mathbb {R}^{m-2}\) .
Let \(t_1,t_2 \in (-T,T)\)  with \(t_1<t_2\)  and let \(\epsilon >0\) . In the similar way in [7]} and in [8]}, Lemma 5.1 we consider the functions \(\xi _1(x,y_1), \dots , \xi _n(x,y_n)\)  and \(h(x)\)  defined for small \(\delta >0\)  by
\(\xi _s (x,y_s)= \left\lbrace \begin{array}{l}\quad \quad 0\hspace{91.04872pt} -T<x<T, \, \, -r_s <y_s\le \gamma _s(x) -\epsilon -\delta \\\frac{1}{\delta }(y_s -\gamma _s(x)+\epsilon +\delta ) \hspace{18.49411pt} -T<x<T, \, \, \gamma _s (x) -\epsilon -\delta <y_s\le \gamma _s (x) -\epsilon \\\quad \quad 1 \hspace{91.04872pt} -T<x<T, \, \, \gamma _s(x) -\epsilon <y_s\le \gamma _s(x)\\\frac{1}{\delta }(-y_s +\gamma _s(x) +\delta ) \hspace{28.45274pt} -T<x<T, \, \, \gamma _s (x) <y_s\le \gamma _s(x) +\delta \\\quad \quad 0 \hspace{91.04872pt} -T<x<T, \, \, \gamma _s(x) +\delta <y_s< r_s\end{array}\right.\) 
for \(s=1,\dots ,n\)  and
\(h(x) = \left\lbrace \begin{array}{l}\quad \quad 0\hspace{71.13188pt} -T< x \le t_1 -\delta \\\frac{1}{\delta }(x-t_1 +\delta ) \hspace{36.98866pt} t_1 -\delta <x\le t_1\\\quad \quad 1 \hspace{72.55461pt} t_1 <x\le t_2\\\frac{1}{\delta }(-x+t_2 +\delta ) \hspace{29.87547pt} t_2<x \le t_2+\delta \\\quad \quad 0 \hspace{73.97733pt}t_2+\delta <x<T\end{array}\right.\) 
If \(\phi \)  is distributional solution of \((D_2^\phi \phi , \dots , D_m^\phi \phi )=w\)  in \(\mathcal {O} \) , for all test function \(\varphi \)  we have
\(\begin{aligned}\int _{-T}^T \int _{-r_1}^{r_1} \dots \int _{-r_n}^{r_n} (\phi (x,\hat{x}_j,y)\varphi _x(x,y) & +f_1(\phi (x,\hat{x}_j,y)) \varphi _{y_1}(x,y)+\dots +f_n(\phi (x,y)) \varphi _{y_n}(x,y)\\& +w_j(x,\hat{x}_j,y)\varphi (x,y))\, dx dy =0\end{aligned}\) 
where
\(\begin{aligned}f_s(\phi (x_j,\hat{x}_j,y))& = \frac{1}{2} \biggl ( b_{j1}^{(s)}\phi ^2(x_j,\hat{x}_j,y)+\phi (x_j,\hat{x}_j,y)\sum _{l=2}^{m}b_{jl}^{(s)}(\hat{x}_j)_l \biggl )\end{aligned}\) 
for all \((x_j,\hat{x}_j,y)\in \mathcal {O} \)  and \(s=1,\dots ,n\) .
We choose the test function \(\varphi (x,y_1,\dots ,y_n )= \xi _1 (x,y_1) \dots \xi _n (x,y_n) h(x)\)  and compute the limit \(\delta \rightarrow 0^+\) :
\(\begin{aligned}\lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \dots \int _{-r_n}^{r_n} (\phi \varphi _x+f_1(\phi ) ( \xi _1)_{y_1} \xi _2 \dots \xi _n h+\dots +f_n(\phi ) \xi _1 \dots (\xi _n)_{y_n}h+w_j\xi _1 \dots \xi _nh)\, dx dy\end{aligned}\) 
For simplicity we consider the case \(n=2\) . We compute the following limit:
\(\begin{aligned}\lim _{\delta \rightarrow 0^+} & \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} \biggl (\phi \biggl ((\xi _1)_x\xi _2 h+\xi _1(\xi _2)_x h+\xi _1\xi _2 h_x\biggl )+f_1(\phi ) ( \xi _1)_{y_1} \xi _2 h \\& \quad +f_2(\phi ) \xi _1(\xi _2)_{y_2}h + w_j\xi _1 \xi _2h\biggl )\, dx dy_1dy_2=: \lim _{\delta \rightarrow 0^+} I_1+I_2+I_3+I_4+I_5+I_6.\end{aligned}\) 
Then
\(\begin{aligned}\lim _{\delta \rightarrow 0^+} I_1& = \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} \phi (\xi _1)_x\xi _2 h\, dx dy_1dy_2\\& =\int _a^b \int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} \dot{\gamma }_1(x) \Bigl (\phi (x,\hat{x}_j,\gamma _1(x), y_2) - \phi (x,\hat{x}_j,\gamma _1 (x)-\epsilon , y_2)\Bigl )\, dxdy_2\\\lim _{\delta \rightarrow 0^+} I_2& = \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} \phi \xi _1(\xi _2)_x h\, dx dy_1dy_2\\& =\int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)} \dot{\gamma }_2(x) \Bigl (\phi (x,\hat{x}_j,y_1, \gamma _2(x)) - \phi (x,\hat{x}_j,y_1, \gamma _2 (x)-\epsilon )\Bigl )\, dxdy_1\\\end{aligned}\) 
\(\begin{aligned}\lim _{\delta \rightarrow 0^+} I_3& = \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} \phi \xi _1\xi _2 h_x\, dx dy_1dy_2\\& = \int _{\gamma _1(t_1)-\epsilon }^{\gamma _1(t_1)} \int _{\gamma _2(t_1)-\epsilon }^{\gamma _2(t_1)} \phi (t_1,\hat{x}_j,y_1, y_2) \, dy_1dy_2 -\int _{\gamma _1(t_2)-\epsilon }^{\gamma _1(t_2)} \int _{\gamma _2(t_2)-\epsilon }^{\gamma _2(t_2)} \phi (t_2,\hat{x}_j,y_1, y_2) \, dy_1dy_2\\\lim _{\delta \rightarrow 0^+} I_4& = \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} f_1(\phi ) ( \xi _1)_{y_1} \xi _2 h\, dx dy_1dy_2\\& = \int _{t_1}^{t_2} \int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} \Bigl ( f_1(\phi (x,\hat{x}_j,\gamma _1 (x)- \epsilon ,y_2))-f_1(\phi (x,\hat{x}_j,\gamma _1 (x) ,y_2)) \Bigl )\, dxdy_2\\\lim _{\delta \rightarrow 0^+} I_5& = \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} f_2(\phi )\xi _1 ( \xi _2)_{y_2} h\, dx dy_1dy_2\\& = \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)} \Bigl ( f_2(\phi (x,y_1, \gamma _2 (x)- \epsilon ))-f_2(\phi (x,y_1, \gamma _2 (x) )) \Bigl )\, dxdy_1\\\lim _{\delta \rightarrow 0^+} I_6 &= \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2}w_j\xi _1 \xi _2h\, dx dy_1dy_2 = \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)}\int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} w_j\, dxdy_1dy_2\end{aligned}\) 
Hence we obtain
\(\begin{aligned}& \int _{\gamma _1(t_2)-\epsilon }^{\gamma _1(t_2)} \int _{\gamma _2(t_2)-\epsilon }^{\gamma _2(t_2)} \phi (t_2,\hat{x}_j,y_1, y_2) \, dy_1dy_2 - \int _{\gamma _1(t_1)-\epsilon }^{\gamma _1(t_1)} \int _{\gamma _2(t_1)-\epsilon }^{\gamma _2(t_1)} \phi (t_1,\hat{x}_j,y_1, y_2) \, dy_1dy_2\\& \quad - \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)}\int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} w_j (x,\hat{x}_j,y_1, y_2) \, dxdy_1dy_2\\& = \int _{t_1}^{t_2} \int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} f_1(\phi (x,\hat{x}_j,\gamma _1 (x)- \epsilon ,y_2))-f_1(\phi (x,\hat{x}_j,\gamma _1 (x) ,y_2))\\& \quad + \dot{\gamma }_1(x) \Bigl (\phi (x,\hat{x}_j,\gamma _1(x), y_2) - \phi (x,\hat{x}_j,\gamma _1 (x)-\epsilon , y_2)\Bigl )\, dx dy_2 \\& \quad + \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)} f_2(\phi (x,\hat{x}_j,y_1, \gamma _2 (x)- \epsilon ))-f_2(\phi (x,\hat{x}_j,y_1, \gamma _2 (x) )) \\&\quad + \dot{\gamma }_2(x) \Bigl (\phi (x,\hat{x}_j,y_1, \gamma _2(x)) - \phi (x,\hat{x}_j,y_1, \gamma _2 (x)-\epsilon )\Bigl )\, dxdy_1\end{aligned}\) 
From (REF ) with \(n=2\)  we deduce that
\(\begin{aligned}& \int _{\gamma _1(t_2)-\epsilon }^{\gamma _1(t_2)} \int _{\gamma _2(t_2)-\epsilon }^{\gamma _2(t_2)} \phi (t_2,\hat{x}_j,y_1, y_2) \, dy_1dy_2 - \int _{\gamma _1(t_1)-\epsilon }^{\gamma _1(t_1)} \int _{\gamma _2(t_1)-\epsilon }^{\gamma _2(t_1)} \phi (t_1,\hat{x}_j,y_1, y_2) \, dy_1dy_2\\& \quad - \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)}\int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} w_j (x,\hat{x}_j,y_1, y_2) \, dxdy_1dy_2\\& = \int _{t_1}^{t_2} \int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} \frac{1}{2}b_{j1}^{(1)}\biggl ( \phi (x,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2)- \phi (x,\hat{x}_j,\gamma _1(x) ,y_2) \biggl ) \biggl ( \phi (x,\hat{x}_j,\hat{x}_j,\gamma _1(x),y_2)+\\& + \phi (x,\hat{x}_j,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2) -2\phi (x,\hat{x}_j,\hat{x}_j,\gamma _1(x),\gamma _2(x)) \biggl ) \, dxdy_2 + \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)} \frac{1}{2}b_{j1}^{(2)}\biggl ( \phi (x,\hat{x}_j,y_1,\gamma _2(x)-\epsilon )\\ & - \phi (x,y_1,\gamma _2(x) ) \biggl ) \biggl ( \phi (x,y_1,\gamma _2(x)) + \phi (x,y_1, \gamma _2(x)-\epsilon ) -2\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) \biggl ) \, dxdy_1\\\end{aligned}\) 
If \(b_{j1}^{(1)}, b_{j1}^{(2)} = 0\)  the integral (REF ) is zero. On the other hand if \(b_{j1}^{(1)} , b_{j1}^{(2)}\ne 0\) , because \(\phi \)  is locally \(1/2 \) -Hölder continuous along the vertical components with Hölder's constant \(C_h>0\)
\(\begin{aligned}&\frac{|\phi (x,\hat{x}_j,\gamma _1(x),y_2)- \phi (x,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2)| }{\sqrt{\epsilon }} \le C_h\\& \frac{|\phi (x,\hat{x}_j,\gamma _1(x),y_2) -\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) |}{\sqrt{\epsilon }} \le C_h\\& \frac{ |\phi (x,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2) -\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) |}{\sqrt{2\epsilon }} \le C_h\end{aligned}\) 
for \(x\in [t_1,t_2]\)  and \(y_2\in (\gamma _2(x)-\epsilon , \gamma _2(x))\) . Hence dividing by \(\epsilon ^2\)  in \((\ref {lipottobre})\)  and getting to the limit to \(\epsilon \rightarrow 0\) , we obtain
\(\begin{aligned}&\lim _{\epsilon \rightarrow 0 } \frac{1}{\epsilon ^2} \int _{t_1}^{t_2} \int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} \frac{1}{2}b_{j1}^{(1)}\biggl ( \phi (x,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2)- \phi (x,\hat{x}_j,\gamma _1(x) ,y_2) \biggl ) \biggl ( \phi (x,\hat{x}_j,\gamma _1(x),y_2)+\\& \quad + \phi (x,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2) -2\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) \biggl ) \, dxdy_2 \\& \le |b_{j1}^{(1)}| \frac{(1+\sqrt{2})C_h^2}{2} (t_2-t_1).\end{aligned}\) 
Moreover
\(\begin{aligned}&\lim _{\epsilon \rightarrow 0 } \frac{1}{\epsilon ^2} \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)} \frac{1}{2}b_{j1}^{(2)}\biggl ( \phi (x,\hat{x}_j,y_1,\gamma _2(x)-\epsilon )- \phi (x,\hat{x}_j,y_1,\gamma _2(x) ) \biggl ) \biggl ( \phi (x,\hat{x}_j,y_1,\gamma _2(x))+\\& \quad + \phi (x,\hat{x}_j,y_1, \gamma _2(x)-\epsilon ) -2\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) \biggl ) \, dxdy_1\\& \le |b_{j1}^{(2)} | \frac{(1+\sqrt{2})C_h^2}{2} (t_2-t_1),\end{aligned}\) 
where in the last inequality we have used the fact
\(\begin{aligned}&\frac{|\phi (x,\hat{x}_j,y_1,\gamma _2(x)-\epsilon )- \phi (x,\hat{x}_j,y_1,\gamma _2(x)) |}{\sqrt{\epsilon }} \le C_h\\& \frac{|\phi (x,\hat{x}_j,y_1,\gamma _2(x)) -\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) |}{\sqrt{\epsilon }} \le C_h\\& \frac{ |\phi (x,\hat{x}_j,y_1,\gamma _2(x)-\epsilon ) -\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) |}{\sqrt{2\epsilon }} \le C_h\end{aligned}\) 
for \(x\in [t_1,t_2]\)  and \(y_1\in (\gamma _1(x)-\epsilon , \gamma _1(x))\) .
Now putting together \((\ref {dise01})\) , \((\ref {dise02})\)  and \((\ref {dise03})\) , dividing by \(\epsilon ^2\)  and getting to the limit to \(\epsilon \rightarrow 0\) , we obtain
\(\phi (t_2,\hat{x}_j , \gamma _j(t_2)) -\phi (t_1,\hat{x}_j , \gamma _j(t_1 )) \le \biggl (\Vert w_j\Vert _{\mathcal {L}^\infty (\mathcal {O} ,\mathbb {R})} + \frac{(1+\sqrt{2})C_h^2}{2} \Big (|b_{j1}^{(1)} |+ |b_{j1}^{(2)}|\Big ) \biggl ) (t_2-t_1)\) 
for \(t_1,t_2 \in (-T,T)\)  with \(t_1<t_2\) .
In the similar way, using again (REF ) and (REF ) we have
\(\phi (t_2,\hat{x}_j , \gamma _j(t_2)) - \phi (t_1,\hat{x}_j , \gamma _j(t_1)) \ge - \biggl (\Vert w_j\Vert _{\mathcal {L}^\infty (\mathcal {O},\mathbb {R})} + \frac{(1+\sqrt{2})C_h^2}{2} \Big (|b_{j1}^{(1)} |+ |b_{j1}^{(2)}|\Big ) \biggl ) (t_2-t_1)\) 
for \(t_1,t_2 \in (-T,T)\)  with \(t_1<t_2\) .
Hence combining (REF ) and (REF ) we get
\(|\phi (t_2,\hat{x}_j , \gamma _j(t_2 )) - \phi (t_1,\hat{x}_j , \gamma _j(t_1)) | \le \biggl (\Vert w_j\Vert _{\mathcal {L}^\infty (\mathcal {O},\mathbb {R})} + \frac{(1+\sqrt{2})C_h^2}{2} \Big (|b_{j1}^{(1)} |+ |b_{j1}^{(2)}|\Big ) \biggl ) (t_2-t_1),\) 
for \(t_1,t_2 \in (-T , T)\)  with \(t_1<t_2\) , i.e. \(\phi \)  is a Lipschitz map along any characteristic line \(\gamma _{j}\)  as desired.
Example 7.1 We consider free step 2 group \(\mathbb {F}_{m,2}\) . As we said in Remark REF , the composition law \((\ref {1})\)  is given by \(\mathcal {B}^{(s)}\equiv \mathcal {B}^{(l,h)}\)  where \(1\le h< l\le m\)  and \(\mathcal {B}^{(l,h)}\)  has entries \(-1\)  in position \((l,h)\) , 1 in position \((h,l)\)  and 0 everywhere else.
For each \(j=2,\dots ,m\)  and \(\hat{x}_j\in \mathbb {R}^{m-2}\)  fixed the characteristic line \(\gamma _{j}=(\gamma _{j1}, \dots , \gamma _{jn}):[-\delta , \delta ]\rightarrow \mathbb {R}^n\)  has the following form:
\(\begin{aligned}\dot{\gamma }_{js_1} (t)&= b_ {j1 }^{(s_1)} \phi (t,\hat{x}_j, \gamma _j(t))= - \phi (t,\hat{x}_j, \gamma _j(t)) \\\dot{\gamma }_{js_2} (t)&= \frac{1}{2}b^{(s_2)}_{jh_2}(\hat{x}_j )_{h_2}= - \frac{1}{2}(\hat{x}_j )_{h_2}\\\dot{\gamma }_{js_3} (t)&= 0\end{aligned}\) 
where \(\mathcal {B}^{(s_i)}\equiv \mathcal {B}^{(l_i,h_i)}\)  is such that

\(s_1\)  is the unique index \((l_1,h_1)=(j,1)\) .

\(s_2\)  is such that \(h_2<l_2=j\)  and \(h_2\ne 1\) .

\(s_3\)  represents the other cases.


Now we are able to show the proof of Theorem REF .
Fix \(b\in \mathcal {O}\) . According to Remark REF , we would like to prove that there exists \(\hat{C}>0\)  such that
\(|\phi (a^{\prime })-\phi (a)| \le \hat{C} \Vert \hat{\phi }(i(a))^{-1}i(a)^{-1}i(a^{\prime })\hat{\phi }(i(a)) \Vert  \qquad \mbox{for all } \, a, a^{\prime } \in \mathcal {U}(b,\delta )\) 
with \(\delta >0\) . Let \(a=( x,y),a^{\prime }= ( x^{\prime },y^{\prime })\)  points of \(\mathcal {O}\)  be sufficiently close to \(b\) , and let \(\bar{D}_j\)  be the vector fields given by \(\bar{D}_j=(x^{\prime }_j-x_j)D^\phi _j\)  for \(j\in \lbrace 2,\dots ,m\rbrace \) . We define
\(\begin{aligned}a_1 & :=a\\a_2 & :=\exp (\bar{D} _2)(a_1)\\a_3 & :=\exp (\bar{D} _3)(a_2)\\\vdots & \\a_m & :=\exp (\bar{D}_m)(a_{m-1}).\end{aligned}\) 
More precisely for \(j\in \lbrace 2,\dots ,m\rbrace \)
\(a_j=(x^{\prime }_2,\dots ,x^{\prime }_j,x_{j+1},\dots ,x_m,y^{a_j})\) 
with
\(\begin{aligned}y^{a_j}_s & = y_s+\sum _{l=2}^{j} \left( b_{l1}^{(s)} \int _0^{x^{\prime }_l-x_l} \phi \Big (\exp (rD^\phi _l (a_{l-1}))\Big )\, dr +\frac{1}{2}(x^{\prime }_l-x_l)\Big (\sum _{ h=2 }^{l} x^{\prime }_h b_{lh}^{(s)}+\sum _{ h=l+1}^{m} x_h b_{lh}^{(s)}\Big ) \right)\\& = y^{a_{j-1}}_s +\left( b_{j1}^{(s)} \int _0^{x^{\prime }_j-x_j} \phi \Big (\exp (rD^\phi _j (a_{j-1}))\Big )\, dr +\frac{1}{2}(x^{\prime }_j-x_j)\Big (\sum _{ h=2 }^{j} x^{\prime }_h b_{jh}^{(s)}+\sum _{ h=j+1}^{m} x_hb_{jh}^{(s)}\Big ) \right)\end{aligned}\) 
for \(s=1,\dots ,n\) . By Lemma REF  \((-\delta ,\delta )\ni r\mapsto \phi \Big (\exp (rD^\phi _j (a_{j-1}))\Big )\)  is Lipschitz for all \(j=2,\dots , m\)  and so \(a_2,\dots ,a_m \)  are well defined if \(a,a^{\prime }\in \mathcal {U}(b,\delta )\)  for a sufficiently small \(\delta \) .
We observe that
\(|\phi (a^{\prime })-\phi (a)| \le |\phi (a^{\prime })-\phi (a_m)| + \sum _{l=2}^{m} |\phi (a_l)-\phi (a_{l-1} )|\) 
and by Lemma \(\ref {lemma5.1bcsc} \)  there is \(C>0\)  such that
\(\sum _{l=2}^{m} |\phi (a_l)-\phi (a_{l-1} )| \le \sum _{l=2}^{m} C | x^{\prime }_l - x_l |_{\mathbb {R}^{m-1}} \le C \Vert \hat{\phi }(i(a))^{-1}i(a)^{-1}i(a^{\prime })\hat{\phi }(i(a)) \Vert .\) 
Hence in order to establish \((\ref {lipsch})\)  we show that there exists \(C_1>0\)  such that
\(|\phi (a^{\prime })-\phi (a_m)|\le C_1 \Vert \hat{\phi }(i(a))^{-1}i(a)^{-1}i(a^{\prime })\hat{\phi }(i(a)) \Vert .\) 
Recalling that \(a_m=(x^{\prime },y^{a_m})\)  and using \(\phi \)  is locally \(1/2 \) -Hölder continuous along the vertical components with Hölder's constant \(C_h>0\) , it follows
\(|\phi (a^{\prime })-\phi (a_m)|\le C_h |y^{\prime }-y^{a_m}|^{1/2}_{\mathbb {R}^{n}}\) 
Moreover arguing as in the proof of the implication \(4. \Rightarrow 2.\)  in Theorem 5.7 in [9]}, we have
\(\begin{aligned}|y^{\prime }-y^{a_m}|_{\mathbb {R}^{n}} & \le \sum _{ s=1}^{n} \biggl |y^{\prime }_s-y_s+\sum _{l=2}^{m} \biggl ( b_{1l}^{(s)} \int _0^{x^{\prime }_l-x_l} \phi \Big (\exp (rD^\phi _l (a_{l-1}))\Big )\, dr +\\&\quad -\frac{1}{2}(x^{\prime }_l-x_l)\Big (\sum _{ i=2 }^{l} x^{\prime }_i b_{li}^{(s)}+\sum _{ i=l+1}^{m} x_i b_{li}^{(s)}\Big ) \biggl ) \biggl | \\& \le \sum _{ s=1}^{n} \biggl |y^{\prime }_s- y_s +\phi (a)\sum _{ l=2 }^{m} (x^{\prime }_l-x_l) b_{1l}^{(s)} -\frac{1}{2}\langle \mathcal {B}^{(s)} x, x^{\prime }- x \rangle \biggl |\\& \quad + \sum _{ s=1}^{n} \biggl | -\frac{1}{2} \sum _{l=2}^{m}(x^{\prime }_l-x_l)\Big (\sum _{ i=2 }^{l} x^{\prime }_i b_{li}^{(s)}+\sum _{ i=l+1}^{m} x_i b_{li}^{(s)}\Big ) + \frac{1}{2} \langle \mathcal {B}^{(s)} x,x^{\prime }- x \rangle \biggl | \\& \quad +\sum _{ s=1}^{n} \biggl | -\phi (a)\sum _{ l=2 }^{m} (x^{\prime }_l-x_l) b_{1l}^{(s)} +\sum _{l=2}^{m} b_{1l}^{(s)} \int _0^{x^{\prime }_l-x_l} \phi \Big (\exp (r D^\phi _l (a_{l-1}))\Big )\, dr \biggl | \\& \le c_1 \Vert \hat{\phi }(i(a))^{-1}i(a)^{-1}i(a^{\prime })\hat{\phi }(i(a)) \Vert ^2 + \frac{1}{2}n\mathcal {B}_M|x^{\prime }-x|^2_{\mathbb {R}^{m-1}} +\\& \quad +\sum _{ s=1}^{n} \biggl | -\phi (a)\sum _{ l=2 }^{m} (x^{\prime }_l-x_l) b_{1l}^{(s)} +\sum _{l=2}^{m} b_{1l}^{(s)} \int _0^{x^{\prime }_l-x_l} \phi \Big (\exp (r D^\phi _l (a_{l-1}))\Big )\, dr \biggl | \\\end{aligned}\) 
where \(c_1 \)  is given by (REF ) and \(\mathcal {B}_{M} = \max \lbrace  b_{ij}^{(s)} \, | \, i,j=1,\dots , m \, , s=1,\dots , n \rbrace \) .
Note that we have used
\(\begin{aligned}\frac{1}{2} \langle \mathcal {B}^{(s)} x, x^{\prime } & - x\rangle -\frac{1}{2} \sum _{l=2}^{m} (x^{\prime }_l -x_l)\Big (\sum _{ i=2 }^{l} x^{\prime }_i b_{li}^{(s)} +\sum _{ i=l+1}^{m} x_i b_{li}^{(s)}\Big ) \\&= -\frac{1}{2} \sum _{l=2}^{m} (x^{\prime }_l-x_l)\Big (\sum _{ i=2 }^{l} x^{\prime }_i b_{li}^{(s)}+\sum _{ i=l+1}^{m} x_i b_{li}^{(s)} -\sum _{ i=2}^{m} x_i b_{li}^{(s)} \Big ) \\& \le \frac{1}{2}n\mathcal {B}_M|x^{\prime }-x|^2_{\mathbb {R}^{m-1}} .\end{aligned}\) 
Finally, the last term
\(\begin{aligned}& \sum _{ s=1}^{n} \biggl | -\phi (a)\sum _{ l=2 }^{m} (x^{\prime }_l-x_l) b_{1l}^{(s)}+\sum _{l=2}^{m} b_{1l}^{(s)} \int _0^{x^{\prime }_l-x_l} \phi \Big (\exp (r D^\phi _l (B_{l-1}))\Big )\, dr \biggl | \\&\qquad \le R_1(a,a^{\prime })+R_2(a,a^{\prime })\end{aligned}\) 
where
\(\begin{aligned}R_1(a,a^{\prime })& := \sum _{ s=1}^{n} \sum _{l=2}^{m}\biggl | b_{1l}^{(s)} \int _0^{x^{\prime }_l-x_l} \phi \Big (\exp (rD^\phi _l (a_{l-1}))\Big )\, dr - b_{1l}^{(s)}\phi (a_{l-1})(x^{\prime }_l-x_l) \biggl |\\R_2(a,a^{\prime }) &:= \sum _{ s=1}^{n} \biggl | \sum _{l=2}^{m} b_{1l}^{(s)}(x^{\prime }_l-x_l) \Big ( \phi (a_{l-1})-\phi (a)\Big ) \biggl | \\\end{aligned}\) 
We would like to show that there exist \(C_1, C_2 >0\)  such that
\(R_1(a,a^{\prime }) \le C_1 |x^{\prime }-x|^2_{\mathbb {R}^{m-1}}\) 
\(R_2(a,a^{\prime }) \le C_2 |x^{\prime }-x|^2_{\mathbb {R}^{m-1}}\) 
for all \(a,a^{\prime }\in \mathcal {U}(b,\delta )\)  and consequently
\(|y^{\prime }-y^{a_m}|_{\mathbb {R}^{n}} \le c_1 \Vert \hat{\phi }(i(a))^{-1}i(a)^{-1}i(a^{\prime })\hat{\phi }(i(a)) \Vert ^2 + \left( \frac{1}{2}n\mathcal {B}_M +C_1+C_2 \right) |x^{\prime }-x|^2_{\mathbb {R}^{m-1}}\) 
Hence there is \( C_3>0\)  such that
\(|y^{\prime }-y^{a_m}|^{1/2} _{\mathbb {R}^{n}} \le C_3 \Vert \hat{\phi }(i(a))^{-1}i(a)^{-1}i(a^{\prime })\hat{\phi }(i(a)) \Vert ,\) 
i.e. (REF ) is true.
We start to consider \(R_1(a,a^{\prime })\) . Fix \(l=2,\dots , m\) . For \(t\in [-\delta , \delta ]\)  we define
\(g_l (t):=\sum _{s=1}^{n} b_{1l}^{(s)} \biggl (\int _0^t \phi (\exp (rD^\phi _l)(a_{l-1}))\, dr - t \phi (a_{l-1}) \biggl )\) 
Observe that
\(\begin{aligned}b_{1l}^{(s)}\int _0^t \left( \phi (\exp (rD^\phi _l)(a_{l-1})) - \phi (a_{l-1}) \right) \, dr = O(t^2)\end{aligned}\) 
and so there is \(C_l>0\)  such that
\(|g_l (t)| \le C_l t^2, \hspace{14.22636pt} \forall t\in [-\delta , \delta ]\) 
Hence set \(t=x^{\prime }_l-x_l\)  we get
\(|g_l (x^{\prime }_l-x_l)| \le C_l (x^{\prime }_l-x_l)^2\) 
and consequently \((\ref {r1})\)  follows from
\(\sum _{l=2}^{m} |g_l (x^{\prime }_l-x_l )| \le \sum _{l=2}^{m} C_l (x^{\prime }_l-x_l)^2 \le C_1|x^{\prime }-x|_{\mathbb {R}^{m-1}} ^2.\) 
Now we consider \(R_2(a,a^{\prime })\) . Observe that
\(\begin{aligned}\sum _{ s=1}^{n} \biggl | \sum _{l=2}^{m} b_{1l}^{(s)}(x^{\prime }_l& -x_l) \Big ( \phi (a_{l-1})-\phi (a)\Big ) \biggl | \le n \mathcal {B}_M \sum _{l=2}^{m} |x^{\prime }_l-x_l| \left| \phi (a_{l-1})-\phi (a) \right| \\& \le n \mathcal {B}_M \sum _{l=2}^{m} |x^{\prime }_l-x_l| \Big (\sum _{h=2}^{l-1}|\phi (a_h)-\phi (a_{h-1})| \Big )\\& \le n \mathcal {B}_M \sum _{l=2}^{m} |x^{\prime }_l-x_l| \Big ( \sum _{h=2}^{l-1}\Big | \int _0^1 (\bar{D}_h \phi )(\exp (r \bar{D}_h (a_{h-1})))\, dr \Big |\Big ) \\& \le n \mathcal {B}_M \sum _{l=2}^{m} |x^{\prime }_l-x_l| \Big (\sum _{h=2}^{l-1} \Big |(x^{\prime }_h-x_h) \left(D^\phi _h \phi ( a_{h-1} )+o(1) \right) \Big | \Big )\\& \le n \mathcal {B}_M C |x^{\prime }-x|^2_{\mathbb {R}^{m-1}}\end{aligned}\) 
Then \((\ref {r2})\)  follows with \( C_2:=n \mathcal {B}_M C \)  and (REF ) is true.
Finally putting together \((\ref {rimanente5})\) , \((\ref {rimanente4})\)  and \((\ref {rimanenteok1})\) , \((\ref {lipsch})\)  holds and the proof is complete.

A characterization of \(\mathbb {G}\) -regular hypersurfaces
In this section we will prove a new characterization of \(\mathbb {G}\) -regular hypersurfaces (see Definition REF ). Here we consider the non linear first order system
\(\left(D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi \right)= w\) 
where \(w:\mathcal {O}\rightarrow \mathbb {R}^{m-1}\)  is a given continuous function and not measurable as opposed to the previous sections.
In [9]}, we give some equivalent conditions about \(\mathbb {G}\) -regular hypersurface inside the Carnot groups of step 2 (see Theorem 5.7 in [9]}). In this section we give another one. More precisely, we show that if a continuous map \(\phi :\mathcal {O}\subset \mathbb {R}^{m+n-1}\rightarrow \mathbb {R}\)  is locally little \(1/2 \) -Hölder continuous along the vertical components, then \(\phi \)  is a distributional solution in an open set \(\mathcal {O}\)  of the non linear first order system \((\ref {sistema})\)  if and only if \(\hat{\phi }: \hat{\mathcal {O}}\rightarrow  is uniformly intrinsic differentiable in \) O\( and consequently its graph is a \) G\(-regular hypersurface. The main equivalence result is contained in Theorem \ref {CharacterizationRegularSurfaces}.\) In order to state the equivalence result we have to be precise about the meaning of being a solution of (REF ). To this aim we recall a notion of generalized solutions of systems of this kind. These generalized solutions, denoted
broad* solutions were introduced and studied for the system (REF ) inside Heisenberg groups in [12]}, [13]}. For a more complete bibliography we refer to the bibliography in [12]}. Our strategy will be to prove that each continuous distributional solution of the system (REF ) is a broad* solution and vice versa (see Proposition REF ).
Definition 8.1 
Let \(\mathcal {O} \subset \mathbb {R}^{m+n-1}\)  be open and \(w:=\left(w_2,\dots ,w_m\right):\mathcal {O} \rightarrow \mathbb {R}^{m-1}\)  a continuous function.
With the notations of Definition REF  we say that \(\phi \in \mathcal {O},\mathbb {R})\)  is a broad* solution in \(\mathcal {O}\)  of the system
\(\left(D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi \right)= w \) 
if for every \(a_0\in \mathcal {O}\)  there are \(0< \delta _2 < \delta _1\)  and \(m-1\)  maps \(\exp _{a_0}(\cdot D^\phi _j)(\cdot )\)
\(\begin{split}\exp _{a_0}(\cdot D^\phi _j)(\cdot ):[-\delta _2,\delta _2]&\times \mathcal {I}(a_0,\delta _2)\rightarrow \mathcal {I}(a_0, \delta _1)\\(t,&a) \mapsto \exp _{a_0}(tD^\phi _j)(a )\end{split}\) 
for \(j=2,\dots , m\) , where \( \mathcal {I}(a_0, \delta ):= \mathcal {U}(a_0,\delta )\cap \mathbb {W}\)  and \(\mathcal {I}(a_0, \delta _1)\subset \mathcal {O}\) . Moreover these maps, called exponential maps of the vector fields \(D^\phi _2,\dots ,D^\phi _m\) , are required to have the following properties
\(t\mapsto \gamma ^j_a (t):=\exp _{a_0} (tD^\phi _j)(B) \in 1 ([-\delta _2,\delta _2], \mathbb {R}^{m+n-1})\) 
for all \(a\in \mathcal {I}(a_0,\delta _2)\)  and
\(\left\lbrace \begin{array}{l}\dot{\gamma }^j_a =D^\phi _j \circ \gamma ^j_a\\\gamma ^j_a (0)=a\end{array}\right.\) 
\(\phi (\gamma ^j_a (t))-\phi (\gamma ^j_a (0))=\int _0^t w_j (\gamma ^j_a (r))\, dr\) 
once more
for all \(a\in \mathcal {I}(a_0,\delta _2)\) .

Proposition 8.1 
Let \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 and \(, \) W\( the complementary subgroups defined in (\ref {5.2.0}). Let \) : O  be a continuous map, where \(\hat{ \mathcal {O}}\)  is an open subset of \(\mathbb {W}\)  and \(\phi :\mathcal {O} \rightarrow \mathbb {R}\)  is the map associated to \(\hat{\phi }\)  as in \((\ref {phipsi})\) . Then the following conditions are equivalent:

\(\phi \)  is locally little \(1/2 \) -Hölder continuous along the vertical components, i.e. that is
\(\phi \in \mathcal {O},\mathbb {R})\)  and for all \(\mathcal {O}^{\prime } \Subset \mathcal {O}\)  and \((x,y), (x,y^{\prime })\in \mathcal {O}^{\prime }\)
\(\lim _{r\rightarrow 0^+} \sup _{0<|y^{\prime }-y|_{\mathbb {R}^{n}} <r} \, \frac{|\phi (x,y^{\prime })-\phi (x,y)|}{|y^{\prime }-y |_{\mathbb {R}^{n}}^{1/2}} =0.\) 
and there exists \(w\in \mathcal {O} , \mathbb {R}^{m-1})\)  such that \(\phi \)  is a broad* solution of
\(\left(D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi \right)= w, \quad \mbox{ in } \mathcal {O}\) 

\(\phi \)  is locally little \(1/2 \) -Hölder continuous along the vertical components and there exists \(w\in \mathcal {O} , \mathbb {R}^{m-1})\)  such that \(\phi \)  is distributional solution of
\(\left(D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi \right)= w, \quad \mbox{ in } \mathcal {O}\) 


\(\mathbf {(1) \Rightarrow (2)}\) .
By Theorem \(5.7\)  in [9]}, we know that there is a family of functions \(\phi _\epsilon \in \mathbb {C}^1(\mathcal {O}, \mathbb {R})\)  such that for all \(\mathcal {O}^{\prime }\Subset \mathcal {O}\) ,
\(\phi _\epsilon \rightarrow \phi \quad and \quad D_j^{{\phi _\epsilon }} {\phi _\epsilon } \rightarrow D_j^{{\phi }} {\phi }\) 
for \(j=2,\dots , m\)  uniformly on \(\mathcal {O} ^{\prime }\)  as \(\epsilon \rightarrow 0^+\) . Then for each \(j=2,\dots , m\) , \(\epsilon >0\)  and \(\zeta \in 1 _c (\mathcal {O}, \mathbb {R})\)
\(\int _\mathcal {O} \phi _\epsilon \left( \,X_j \zeta +\phi _\epsilon \sum _{s=1}^n b_ {j1 }^{(s)} Y_s \zeta \right) \, d\mathcal {L}^{m+n-1} =- \int _\mathcal {O} D^{\phi _\epsilon }_j \phi _\epsilon \, \zeta \, d \mathcal {L}^{m+n-1}\) 
Using (REF ) and getting to the limit for \(\epsilon \rightarrow 0^+\)  we have that
\(\int _\mathcal {O} \phi \left( \,X_j \zeta +\phi \sum _{s=1}^n b_ {j1 }^{(s)} Y_s \zeta \right) \, d\mathcal {L}^{m+n-1} =- \int _\mathcal {O} D^\phi _j \phi \, \zeta \, d \mathcal {L}^{m+n-1}\) 
Hence \(\phi \)  is distributional solution of \(\left(D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi \right)= w\)  in \(\mathcal {O} \) .
\(\mathbf {(2) \Rightarrow (1)}\) .
Let \(a_0\in \mathcal {O} \) , \(\delta _1>0\)  and \(\mathcal {I}:= \mathcal {U} (a_0,\delta _1) \cap \mathbb {W}\)  and \(\mathcal {I}(a_0, \delta _1)\subset \mathcal {O}\) . Denote
\(K:= \sup _{(x,y)\in \mathcal {I}} \sum _{l=2}^{m}|x_l|,\quad M:=\Vert \phi \Vert _{\mathcal {L}^\infty (\mathcal {I} ,\mathbb {R})}, \quad \delta _2 <\frac{\delta _1}{ 2+\frac{1}{2} K\mathcal {B}_M+M\mathcal {B}_M},\) 
where \(\mathcal {B}_{M} = \max \lbrace  b_{jl}^{(s)} \, | \, l,j=1,\dots , m \, , s=1,\dots , n \rbrace \) .
Peano's Theorem yields that for all \(a=(x_j,\hat{x}_j,y)\in \overline{ \mathcal {I} (a_0,\delta _2)}\)  there exists 1 function \( (\gamma _{j1},\dots , \gamma _{jn}) :[-\delta _2,\delta _2]\rightarrow \mathbb {R}^n\)  such that
\(\gamma ^j_{a}(t)=(x_j+t,\hat{x}_j , \gamma _{j1} (t),\dots , \gamma _{jn}(t))\in \mathcal {I} \quad \mbox{for } t\in [-\delta _2,\delta _2],\) 
and \( \gamma _{js}(t)\)  is a solution of the Cauchy problem
\(\left\lbrace \begin{array}{l}\dot{\gamma }_{js}(t)=\frac{1}{2} \sum _{l=2}^{m}(\hat{x}_j)_l b_{jl}^{(s)}+b_{j1}^{(s)} \phi (\gamma _a^j (t)), \quad \mbox{ for } t\in [-\delta _2,\delta _2]\\\\\gamma _{js}(0)=y_s, \\\end{array}\right.\) 
for \(s=1,\dots ,n\) . It is clear that if \(b_ {j1 }^{(s)}= 0\) , then \( \gamma _{js}(t)=y_s+\frac{1}{2}t \sum _{l=2}^{m}(\hat{x}_j)_l b_{jl}^{(s)}\)  for \(t\in [-\delta _2,\delta _2]\) . On the other hand, if \(b_ {j1 }^{(s)}\ne 0\) , then the map \(\phi (\gamma _a^j (\cdot ))\)  satisfies the following ODE
\(\frac{d}{dt} \left(b_ {j1 }^{(s)} \phi (\gamma _a^j (t)) \right) = b_ {j1 }^{(s)} w_{j} (\gamma _a^j (t)) \\\) 
with \(t\in [-\delta , \delta ]\)  for some \(\delta >0\) . Indeed, we can repeat verbatim the proof of Lemma REF  and so we obtain (REF ) with \(b_{j1}^{(s)}\ne 0\) . Moreover because \(\phi \)  is locally little \(1/2 \) -Hölder continuous along the vertical components, upon dividing (REF ) by \(\epsilon ^2\)  and getting to the limit to \(\epsilon \rightarrow 0\) , we have that
\(\begin{aligned}&\phi (\gamma _a^j (t_2)) - \phi (\gamma _a^j (t_1)) - \int _{t_1}^{t_2} w_j (\gamma _a^j (t)) \, dt = 0 \\\end{aligned}\) 
for \(t_1,t_2 \in [-\delta ,\delta ]\)  with \(\delta >0\)  and \(t_1<t_2\) .
In particular \(\phi (\gamma _a^j (\cdot )) \)  and \(\dot{\gamma }_{js} (\cdot )\)  are \(1([-\delta _3,\delta _3] , \mathbb {R})\)  for \(s=1,\dots , n\)  where \(\delta _3:=\min \lbrace  \delta , \delta _2\rbrace  \) . Therefore the curve \(\gamma ^j_{a}:[-\delta _3,\delta _3] \rightarrow \mathcal {I}\)  satisfies the conditions of the Definition \(\ref {defbroad*}\)  for each \(a\in \mathcal {I} (a_0,\delta _3) := \mathcal {U} (a_0,\delta _3) \cap \mathbb {W}\)  and \(\mathcal {I}(a_0, \delta _3)\subset \mathcal {O}\) .
Then, for each \(j=2,\dots , m\) , \(\exp _{a_0}(\cdot D^\phi _j)(\cdot ):[-\delta _3,\delta _3]\times \mathcal {I} (a_0,\delta _3) \rightarrow \mathcal {I}\)  defined as \(\exp _{a_0}(t D^\phi _{j})(a):= \gamma ^j_{a}(t)\)  is a family of exponential maps at \(a_0\)  which we were looking for. This completes the proof of the implication \((4)\Rightarrow (5)\) .
By Proposition REF  and Theorem \(5.7\)  in [9]}, it immediately follows
Theorem 8.2 
Let \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 and \(, \) W\( the complementary subgroups defined in (\ref {5.2.0}). Let \) : O  be a continuous map, where \(\hat{ \mathcal {O}}\)  is an open subset of \(\mathbb {W}\)  and \(\phi :\mathcal {O} \rightarrow \mathbb {R}\)  is the map associated to \(\hat{\phi }\)  as in \((\ref {phipsi})\) . Then the following conditions are equivalent:

\(\mathrm {graph}\,(\hat{\phi }) \)  is a \(\mathbb {G}\) -regular hypersurface and for all \(a\in \mathrm {graph}\,(\hat{\phi })\)  there is \(r=r(a)>0\)  and \(f\in \mathbb {C}_{\mathbb {G}}^1(\mathcal {U}(a,r)),\mathbb {R})\)  with \(X_1f>0\)  such that
\(\mathrm {graph}\,(\hat{\phi })\cap \mathcal {U}(a,r)=\lbrace  p:f(p)=0\rbrace \) 

\(\hat{\phi }\)  is u.i.d. in \( \hat{\mathcal {O}}\) .

\(D_j^{\phi }\phi \)  interpreted in distributional sense is a continuos function in \(\mathcal {O}\)  and for \(0<\epsilon <1\)  there is a family of functions \(\phi _\epsilon \in \mathbb {C}^1(\mathcal {O}, \mathbb {R})\)  such that for all \(\mathcal {O}^{\prime }\Subset \mathcal {O}\) ,
\(\phi _\epsilon \rightarrow \phi \quad and \quad D_j^{{\phi _\epsilon }} {\phi _\epsilon } \rightarrow D_j^{{\phi }} {\phi }\) 
for \(j=2,\dots , m\) , uniformly on \(\mathcal {O} ^{\prime }\)  as \(\epsilon \rightarrow 0^+\) .

\(\phi \)  is locally little \(1/2 \) -Hölder continuous along the vertical components and there exists \(w\in \mathcal {O} , \mathbb {R}^{m-1})\)  such that \(\phi \)  is a broad* solution of
\(\left(D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi \right)= w, \quad \mbox{ in } \mathcal {O}\) 

\(\phi \)  is locally little \(1/2 \) -Hölder continuous along the vertical components and there exists \(w\in \mathcal {O} , \mathbb {R}^{m-1})\)  such that \(\phi \)  is distributional solution of
\(\left(D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi \right)= w, \quad \mbox{ in } \mathcal {O}\) 


Remark 8.3 We recall that in Theorem \(5.7\)  in [9]}, we consider a locally little \(1/2 \) -Hölder continuous map but in the proof we just use that \(\phi \)  is locally little \(1/2 \) -Hölder continuous along the vertical components.

